Chapter 4: Trees

Tree Fundamentals

A tree is a fundamental hierarchical data structure consisting of nodes connected by edges. Unlike linear data structures (arrays, linked lists), trees represent relationships in a parent-child hierarchy, making them ideal for representing hierarchical data, organizing information, and implementing efficient search and retrieval operations.

Key Terminology

Node: A fundamental unit containing data and references to child nodes
Root: The topmost node in a tree (has no parent)
Parent: A node that has child nodes
Child: A node directly connected to a parent node
Leaf: A node with no children (also called external node)
Internal Node: A node with at least one child
Edge: Connection between two nodes
Path: Sequence of nodes connected by edges
Depth: Number of edges from root to a node
Height: Maximum depth of any node in the tree (or depth of deepest node)
Level: All nodes at the same depth
Subtree: A tree formed by a node and all its descendants
Ancestor: Any node on the path from root to a given node
Descendant: Any node in the subtree rooted at a given node
Sibling: Nodes with the same parent

Tree Properties

A tree with n nodes has exactly n-1 edges
There is exactly one path between any two nodes
Adding any edge creates a cycle (no longer a tree)
Removing any edge disconnects the tree
A tree is a connected acyclic graph

Basic Tree Implementation

class TreeNode:
    def __init__(self, val=0, children=None):
        self.val = val
        self.children = children if children is not None else []

# Example: Creating a tree
root = TreeNode(1)
root.children = [TreeNode(2), TreeNode(3), TreeNode(4)]
root.children[0].children = [TreeNode(5), TreeNode(6)]

# Tree structure:
#       1
#     / | \
#    2  3  4
#   / \
#  5   6

Tree Height and Depth Calculation

def tree_height(root):
    """Calculate the height of a tree (maximum depth)"""
    if not root:
        return -1  # Empty tree has height -1, single node has height 0
    
    if not root.children:  # Leaf node
        return 0
    
    # Height is 1 + maximum height of all subtrees
    return 1 + max(tree_height(child) for child in root.children)

def node_depth(root, target, depth=0):
    """Find the depth of a specific node"""
    if not root:
        return -1
    
    if root.val == target:
        return depth
    
    # Search in all children
    for child in root.children:
        result = node_depth(child, target, depth + 1)
        if result != -1:
            return result
    
    return -1  # Node not found

Types of Trees

General Tree: No restriction on number of children per node
Binary Tree: Each node has at most 2 children
Binary Search Tree (BST): Binary tree with ordering property
AVL Tree: Self-balancing BST
Red-Black Tree: Self-balancing BST with color properties
Heap: Complete binary tree with heap property
Trie: Prefix tree for string operations
B-Tree: Multi-way search tree for databases

Binary Trees

A binary tree is a tree data structure where each node has at most two children, referred to as the left child and right child. Binary trees are fundamental in computer science and form the basis for many advanced data structures like binary search trees, heaps, and expression trees.

Binary Tree Structure

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

# Example binary tree:
#        1
#       / \
#      2   3
#     / \   \
#    4   5   6

Types of Binary Trees

1. Full Binary Tree

Every node has either 0 or 2 children (no node has exactly 1 child).

# Full binary tree example:
#        1
#       / \
#      2   3
#     / \
#    4   5

2. Complete Binary Tree

All levels are completely filled except possibly the last level, which is filled from left to right. Used in heaps.

# Complete binary tree example:
#        1
#       / \
#      2   3
#     / \ /
#    4  5 6

3. Perfect Binary Tree

All internal nodes have exactly 2 children, and all leaves are at the same level. A perfect binary tree of height h has 2^h+1 - 1 nodes.

# Perfect binary tree example:
#        1
#       / \
#      2   3
#     / \ / \
#    4  5 6  7

4. Balanced Binary Tree

The height difference between left and right subtrees of any node is at most 1. Ensures O(log n) operations.

5. Degenerate (Pathological) Tree

Each parent node has only one child, essentially a linked list. Worst case for tree operations (O(n)).

# Degenerate tree (worst case):
#    1
#     \
#      2
#       \
#        3
#         \
#          4

Binary Tree Properties

Maximum number of nodes at level l: 2^l
Maximum number of nodes in a binary tree of height h: 2^h+1 - 1
Minimum number of nodes in a binary tree of height h: h + 1
Number of leaf nodes in a full binary tree: (n + 1) / 2
Number of internal nodes in a full binary tree: (n - 1) / 2

Common Binary Tree Operations

Count Nodes

def count_nodes(root):
    """Count total number of nodes in binary tree"""
    if not root:
        return 0
    return 1 + count_nodes(root.left) + count_nodes(root.right)

Count Leaf Nodes

def count_leaves(root):
    """Count number of leaf nodes"""
    if not root:
        return 0
    if not root.left and not root.right:
        return 1
    return count_leaves(root.left) + count_leaves(root.right)

Maximum Depth/Height

def max_depth(root):
    """Calculate maximum depth of binary tree"""
    if not root:
        return 0
    return 1 + max(max_depth(root.left), max_depth(root.right))

Check if Balanced

def is_balanced(root):
    """Check if binary tree is balanced"""
    def check_balance(node):
        if not node:
            return 0, True
        
        left_height, left_balanced = check_balance(node.left)
        right_height, right_balanced = check_balance(node.right)
        
        height = 1 + max(left_height, right_height)
        balanced = (left_balanced and right_balanced and 
                   abs(left_height - right_height) <= 1)
        
        return height, balanced
    
    _, balanced = check_balance(root)
    return balanced

Check if Same Tree

def is_same_tree(p, q):
    """Check if two binary trees are identical"""
    if not p and not q:
        return True
    if not p or not q:
        return False
    return (p.val == q.val and 
            is_same_tree(p.left, q.left) and 
            is_same_tree(p.right, q.right))

Check if Symmetric

def is_symmetric(root):
    """Check if binary tree is symmetric (mirror image)"""
    def is_mirror(left, right):
        if not left and not right:
            return True
        if not left or not right:
            return False
        return (left.val == right.val and 
                is_mirror(left.left, right.right) and 
                is_mirror(left.right, right.left))
    
    if not root:
        return True
    return is_mirror(root.left, root.right)

Invert Binary Tree

def invert_tree(root):
    """Invert (mirror) a binary tree"""
    if not root:
        return None
    
    # Swap left and right subtrees
    root.left, root.right = root.right, root.left
    
    # Recursively invert subtrees
    invert_tree(root.left)
    invert_tree(root.right)
    
    return root

Binary Tree Construction

From Array (Level-order)

def build_tree_from_array(arr):
    """Build binary tree from level-order array representation"""
    if not arr or arr[0] is None:
        return None
    
    root = TreeNode(arr[0])
    queue = [root]
    i = 1
    
    while queue and i < len(arr):
        node = queue.pop(0)
        
        if i < len(arr) and arr[i] is not None:
            node.left = TreeNode(arr[i])
            queue.append(node.left)
        i += 1
        
        if i < len(arr) and arr[i] is not None:
            node.right = TreeNode(arr[i])
            queue.append(node.right)
        i += 1
    
    return root

From Preorder and Inorder

def build_tree_pre_inorder(preorder, inorder):
    """Build binary tree from preorder and inorder traversals"""
    if not preorder or not inorder:
        return None
    
    root_val = preorder[0]
    root = TreeNode(root_val)
    root_index = inorder.index(root_val)
    
    root.left = build_tree_pre_inorder(
        preorder[1:root_index + 1], 
        inorder[:root_index]
    )
    root.right = build_tree_pre_inorder(
        preorder[root_index + 1:], 
        inorder[root_index + 1:]
    )
    
    return root

Tree Traversals

Tree traversal is the process of visiting all nodes in a tree exactly once. Different traversal orders serve different purposes: inorder for BST gives sorted order, preorder for copying trees, postorder for deleting trees, and level-order for breadth-first processing.

1. Inorder Traversal (Left → Root → Right)

Visit left subtree, then root, then right subtree. For BST, this produces values in sorted order.

Recursive Implementation

def inorder_traversal(root):
    """Inorder traversal - returns list of values"""
    result = []
    
    def inorder(node):
        if node:
            inorder(node.left)      # Left
            result.append(node.val) # Root
            inorder(node.right)     # Right
    
    inorder(root)
    return result

# Example tree:
#        1
#       / \
#      2   3
#     / \
#    4   5
# Result: [4, 2, 5, 1, 3]

Iterative Implementation

def inorder_iterative(root):
    """Iterative inorder using stack"""
    result = []
    stack = []
    current = root
    
    while stack or current:
        # Go to leftmost node
        while current:
            stack.append(current)
            current = current.left
        
        # Process node
        current = stack.pop()
        result.append(current.val)
        
        # Move to right subtree
        current = current.right
    
    return result

2. Preorder Traversal (Root → Left → Right)

Visit root first, then left subtree, then right subtree. Useful for copying trees, prefix expressions, and serialization.

Recursive Implementation

def preorder_traversal(root):
    """Preorder traversal - returns list of values"""
    result = []
    
    def preorder(node):
        if node:
            result.append(node.val) # Root
            preorder(node.left)     # Left
            preorder(node.right)    # Right
    
    preorder(root)
    return result

# Example tree:
#        1
#       / \
#      2   3
#     / \
#    4   5
# Result: [1, 2, 4, 5, 3]

Iterative Implementation

def preorder_iterative(root):
    """Iterative preorder using stack"""
    if not root:
        return []
    
    result = []
    stack = [root]
    
    while stack:
        node = stack.pop()
        result.append(node.val)
        
        # Push right first, then left (so left is processed first)
        if node.right:
            stack.append(node.right)
        if node.left:
            stack.append(node.left)
    
    return result

3. Postorder Traversal (Left → Right → Root)

Visit left subtree, then right subtree, then root. Useful for deleting trees, postfix expressions, and calculating directory sizes.

Recursive Implementation

def postorder_traversal(root):
    """Postorder traversal - returns list of values"""
    result = []
    
    def postorder(node):
        if node:
            postorder(node.left)    # Left
            postorder(node.right)   # Right
            result.append(node.val) # Root
    
    postorder(root)
    return result

# Example tree:
#        1
#       / \
#      2   3
#     / \
#    4   5
# Result: [4, 5, 2, 3, 1]

Iterative Implementation

def postorder_iterative(root):
    """Iterative postorder using two stacks"""
    if not root:
        return []
    
    stack1 = [root]
    stack2 = []
    
    while stack1:
        node = stack1.pop()
        stack2.append(node)
        
        if node.left:
            stack1.append(node.left)
        if node.right:
            stack1.append(node.right)
    
    return [node.val for node in reversed(stack2)]

4. Level-Order Traversal (BFS)

Visit nodes level by level from top to bottom, left to right. Uses a queue for breadth-first search. See Chapter 7: Graph Algorithms for BFS details.

Implementation

def level_order_traversal(root):
    """Level-order (BFS) traversal"""
    if not root:
        return []
    
    result = []
    queue = [root]
    
    while queue:
        level_size = len(queue)
        level = []
        
        for _ in range(level_size):
            node = queue.pop(0)
            level.append(node.val)
            
            if node.left:
                queue.append(node.left)
            if node.right:
                queue.append(node.right)
        
        result.append(level)
    
    return result

# Example tree:
#        1
#       / \
#      2   3
#     / \
#    4   5
# Result: [[1], [2, 3], [4, 5]]

5. Zigzag Level-Order Traversal

Level-order traversal but alternating direction at each level.

def zigzag_level_order(root):
    """Zigzag level-order traversal"""
    if not root:
        return []
    
    result = []
    queue = [root]
    left_to_right = True
    
    while queue:
        level_size = len(queue)
        level = []
        
        for _ in range(level_size):
            node = queue.pop(0)
            level.append(node.val)
            
            if node.left:
                queue.append(node.left)
            if node.right:
                queue.append(node.right)
        
        if not left_to_right:
            level.reverse()
        result.append(level)
        left_to_right = not left_to_right
    
    return result

6. Morris Traversal (Space-Optimized)

Inorder traversal using O(1) extra space by temporarily modifying the tree structure using threaded binary trees.

def morris_inorder(root):
    """Morris inorder traversal - O(1) space"""
    result = []
    current = root
    
    while current:
        if not current.left:
            # No left child, visit current and go right
            result.append(current.val)
            current = current.right
        else:
            # Find inorder predecessor
            predecessor = current.left
            while predecessor.right and predecessor.right != current:
                predecessor = predecessor.right
            
            if not predecessor.right:
                # Make current the right child of predecessor
                predecessor.right = current
                current = current.left
            else:
                # Restore tree structure
                predecessor.right = None
                result.append(current.val)
                current = current.right
    
    return result

Traversal Use Cases

Traversal	Use Cases
Inorder	BST sorted output, expression evaluation (infix)
Preorder	Tree copying, prefix expressions, serialization
Postorder	Tree deletion, postfix expressions, directory size calculation
Level-order	BFS, printing tree structure, finding level-specific nodes

Binary Search Trees

A Binary Search Tree (BST) is a binary tree with the ordering property: for any node, all values in the left subtree are less than the node's value, and all values in the right subtree are greater than the node's value. This property enables efficient search, insertion, and deletion operations with average time complexity O(log n).

BST Properties

Ordering Property: left child < parent < right child (for all nodes)
Inorder Traversal: Produces values in sorted order
Uniqueness: Typically, no duplicate values (or handle with count/frequency)
Search Efficiency: O(log n) average, O(n) worst case (degenerate tree)

BST Implementation

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

class BST:
    def __init__(self):
        self.root = None

Search Operation

Search for a value in BST. Compare with root, go left if smaller, right if larger.

def search(root, val):
    """Search for value in BST - recursive"""
    if not root or root.val == val:
        return root
    
    if val < root.val:
        return search(root.left, val)
    else:
        return search(root.right, val)

def search_iterative(root, val):
    """Search for value in BST - iterative"""
    current = root
    while current:
        if current.val == val:
            return current
        elif val < current.val:
            current = current.left
        else:
            current = current.right
    return None

# Time Complexity: O(log n) average, O(n) worst
# Space Complexity: O(log n) recursive, O(1) iterative

Insert Operation

Insert a new value while maintaining BST property.

def insert(root, val):
    """Insert value into BST - recursive"""
    if not root:
        return TreeNode(val)
    
    if val < root.val:
        root.left = insert(root.left, val)
    elif val > root.val:
        root.right = insert(root.right, val)
    # If val == root.val, do nothing (or handle duplicates)
    
    return root

def insert_iterative(root, val):
    """Insert value into BST - iterative"""
    if not root:
        return TreeNode(val)
    
    current = root
    while True:
        if val < current.val:
            if not current.left:
                current.left = TreeNode(val)
                break
            current = current.left
        elif val > current.val:
            if not current.right:
                current.right = TreeNode(val)
                break
            current = current.right
        else:
            break  # Value already exists
    
    return root

# Time Complexity: O(log n) average, O(n) worst

Delete Operation

Delete a node from BST. Three cases: node with no children, one child, or two children.

def delete_node(root, val):
    """Delete node with given value from BST"""
    if not root:
        return root
    
    if val < root.val:
        root.left = delete_node(root.left, val)
    elif val > root.val:
        root.right = delete_node(root.right, val)
    else:
        # Node to delete found
        if not root.left:
            return root.right
        elif not root.right:
            return root.left
        else:
            # Node has two children
            # Find inorder successor (smallest in right subtree)
            successor = find_min(root.right)
            root.val = successor.val
            root.right = delete_node(root.right, successor.val)
    
    return root

def find_min(root):
    """Find node with minimum value in BST"""
    while root.left:
        root = root.left
    return root

# Time Complexity: O(log n) average, O(n) worst

BST Validation

Check if a binary tree is a valid BST.

def is_valid_bst(root):
    """Check if binary tree is valid BST"""
    def validate(node, min_val, max_val):
        if not node:
            return True
        
        if node.val <= min_val or node.val >= max_val:
            return False
        
        return (validate(node.left, min_val, node.val) and 
                validate(node.right, node.val, max_val))
    
    return validate(root, float('-inf'), float('inf'))

# Alternative: Using inorder traversal
def is_valid_bst_inorder(root):
    """Validate BST using inorder traversal property"""
    prev = None
    
    def inorder(node):
        nonlocal prev
        if not node:
            return True
        
        if not inorder(node.left):
            return False
        
        if prev is not None and node.val <= prev:
            return False
        
        prev = node.val
        return inorder(node.right)
    
    return inorder(root)

Find Minimum and Maximum

def find_min_bst(root):
    """Find minimum value in BST"""
    if not root:
        return None
    while root.left:
        root = root.left
    return root.val

def find_max_bst(root):
    """Find maximum value in BST"""
    if not root:
        return None
    while root.right:
        root = root.right
    return root.val

Find Kth Smallest Element

def kth_smallest(root, k):
    """Find kth smallest element in BST using inorder"""
    stack = []
    current = root
    
    while stack or current:
        while current:
            stack.append(current)
            current = current.left
        
        current = stack.pop()
        k -= 1
        if k == 0:
            return current.val
        
        current = current.right
    
    return None

Lowest Common Ancestor (LCA) in BST

def lca_bst(root, p, q):
    """Find lowest common ancestor in BST"""
    if not root:
        return None
    
    # Both values are in left subtree
    if p < root.val and q < root.val:
        return lca_bst(root.left, p, q)
    
    # Both values are in right subtree
    if p > root.val and q > root.val:
        return lca_bst(root.right, p, q)
    
    # Current node is LCA
    return root

Range Sum Query

def range_sum_bst(root, low, high):
    """Sum all values in BST within range [low, high]"""
    if not root:
        return 0
    
    if root.val < low:
        return range_sum_bst(root.right, low, high)
    if root.val > high:
        return range_sum_bst(root.left, low, high)
    
    return (root.val + 
            range_sum_bst(root.left, low, high) + 
            range_sum_bst(root.right, low, high))

Convert Sorted Array to BST

def sorted_array_to_bst(nums):
    """Convert sorted array to height-balanced BST"""
    if not nums:
        return None
    
    mid = len(nums) // 2
    root = TreeNode(nums[mid])
    root.left = sorted_array_to_bst(nums[:mid])
    root.right = sorted_array_to_bst(nums[mid + 1:])
    
    return root

BST Complexity Analysis

Operation	Average Case	Worst Case
Search	O(log n)	O(n)
Insert	O(log n)	O(n)
Delete	O(log n)	O(n)
Space	O(n)	O(n)

Self-Balancing BSTs

To guarantee O(log n) operations, use self-balancing BSTs:

AVL Tree: Maintains balance using height difference (rotation operations)
Red-Black Tree: Maintains balance using color properties (used in many standard libraries)
Splay Tree: Recently accessed elements move to root
Treap: Combines BST and heap properties

Advanced Tree Structures

Trie (Prefix Tree)

A trie is a tree-like data structure for storing strings. Each node represents a character, and paths from root to leaf represent words. Excellent for prefix matching and autocomplete.

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = False

class Trie:
    def __init__(self):
        self.root = TrieNode()
    
    def insert(self, word):
        """Insert word into trie"""
        node = self.root
        for char in word:
            if char not in node.children:
                node.children[char] = TrieNode()
            node = node.children[char]
        node.is_end = True
    
    def search(self, word):
        """Search for complete word"""
        node = self.root
        for char in word:
            if char not in node.children:
                return False
            node = node.children[char]
        return node.is_end
    
    def starts_with(self, prefix):
        """Check if any word starts with prefix"""
        node = self.root
        for char in prefix:
            if char not in node.children:
                return False
            node = node.children[char]
        return True

# Time Complexity: O(m) where m is word length
# Space Complexity: O(ALPHABET_SIZE * N * M) where N is number of words

Segment Tree

A segment tree is used for range queries and range updates. Each node stores information about a segment of the array.

class SegmentTree:
    def __init__(self, arr):
        self.n = len(arr)
        self.size = 1
        while self.size < self.n:
            self.size *= 2
        self.tree = [0] * (2 * self.size)
        self.build(arr)
    
    def build(self, arr):
        """Build segment tree from array"""
        for i in range(self.n):
            self.tree[self.size + i] = arr[i]
        for i in range(self.size - 1, 0, -1):
            self.tree[i] = self.tree[2 * i] + self.tree[2 * i + 1]
    
    def query(self, l, r):
        """Query sum in range [l, r)"""
        l += self.size
        r += self.size
        result = 0
        while l < r:
            if l % 2 == 1:
                result += self.tree[l]
                l += 1
            if r % 2 == 1:
                r -= 1
                result += self.tree[r]
            l //= 2
            r //= 2
        return result
    
    def update(self, index, value):
        """Update value at index"""
        index += self.size
        self.tree[index] = value
        index //= 2
        while index >= 1:
            self.tree[index] = self.tree[2 * index] + self.tree[2 * index + 1]
            index //= 2

Common Tree Problems and Patterns

1. Maximum Path Sum

def max_path_sum(root):
    """Find maximum path sum in binary tree"""
    max_sum = float('-inf')
    
    def dfs(node):
        nonlocal max_sum
        if not node:
            return 0
        
        # Max path from left and right (can be negative)
        left_max = max(0, dfs(node.left))
        right_max = max(0, dfs(node.right))
        
        # Current path sum
        current_sum = node.val + left_max + right_max
        max_sum = max(max_sum, current_sum)
        
        # Return max path from this node (can only use one branch)
        return node.val + max(left_max, right_max)
    
    dfs(root)
    return max_sum

2. Diameter of Binary Tree

def diameter_of_binary_tree(root):
    """Find diameter (longest path between any two nodes)"""
    diameter = 0
    
    def height(node):
        nonlocal diameter
        if not node:
            return 0
        
        left_height = height(node.left)
        right_height = height(node.right)
        
        # Diameter is max of: current diameter, path through current node
        diameter = max(diameter, left_height + right_height)
        
        return 1 + max(left_height, right_height)
    
    height(root)
    return diameter

3. Path Sum Problems

def has_path_sum(root, target_sum):
    """Check if path from root to leaf sums to target"""
    if not root:
        return False
    
    if not root.left and not root.right:
        return root.val == target_sum
    
    return (has_path_sum(root.left, target_sum - root.val) or 
            has_path_sum(root.right, target_sum - root.val))

def path_sum_ii(root, target_sum):
    """Find all paths from root to leaf that sum to target"""
    result = []
    
    def dfs(node, remaining, path):
        if not node:
            return
        
        path.append(node.val)
        
        if not node.left and not node.right and remaining == node.val:
            result.append(path[:])
        else:
            dfs(node.left, remaining - node.val, path)
            dfs(node.right, remaining - node.val, path)
        
        path.pop()
    
    dfs(root, target_sum, [])
    return result

4. Serialize and Deserialize Binary Tree

def serialize(root):
    """Serialize binary tree to string"""
    if not root:
        return "None,"
    return str(root.val) + "," + serialize(root.left) + serialize(root.right)

def deserialize(data):
    """Deserialize string to binary tree"""
    def build_tree(values):
        val = next(values)
        if val == "None":
            return None
        node = TreeNode(int(val))
        node.left = build_tree(values)
        node.right = build_tree(values)
        return node
    
    values = iter(data.split(","))
    return build_tree(values)

5. Binary Tree Right Side View

def right_side_view(root):
    """Return values visible from right side of tree"""
    if not root:
        return []
    
    result = []
    queue = [root]
    
    while queue:
        level_size = len(queue)
        for i in range(level_size):
            node = queue.pop(0)
            if i == level_size - 1:  # Last node in level
                result.append(node.val)
            
            if node.left:
                queue.append(node.left)
            if node.right:
                queue.append(node.right)
    
    return result

Practice Problems

Here are recommended problems to practice tree algorithms:

Easy

LeetCode 104: Maximum Depth of Binary Tree
LeetCode 226: Invert Binary Tree
LeetCode 101: Symmetric Tree
LeetCode 94: Binary Tree Inorder Traversal
LeetCode 108: Convert Sorted Array to BST
LeetCode 112: Path Sum

Medium

LeetCode 98: Validate Binary Search Tree
LeetCode 102: Binary Tree Level Order Traversal
LeetCode 105: Construct Binary Tree from Preorder and Inorder
LeetCode 236: Lowest Common Ancestor of Binary Tree
LeetCode 230: Kth Smallest Element in BST
LeetCode 199: Binary Tree Right Side View
LeetCode 114: Flatten Binary Tree to Linked List
LeetCode 543: Diameter of Binary Tree

Hard

LeetCode 124: Binary Tree Maximum Path Sum
LeetCode 297: Serialize and Deserialize Binary Tree
LeetCode 428: Serialize and Deserialize N-ary Tree
LeetCode 208: Implement Trie (Prefix Tree)
LeetCode 212: Word Search II (Trie)

Summary

Trees are fundamental hierarchical data structures that enable efficient organization and retrieval of data. Binary trees, with their recursive structure, form the basis for many advanced data structures and algorithms.

Key takeaways:

Binary trees have at most two children per node, enabling efficient recursive algorithms
Tree traversals (inorder, preorder, postorder, level-order) serve different purposes
Binary Search Trees maintain ordering property for O(log n) average operations
Self-balancing trees (AVL, Red-Black) guarantee O(log n) worst-case performance
Specialized trees (Trie, Segment Tree) solve domain-specific problems efficiently
Many tree problems follow recursive patterns with base cases and recursive cases

Understanding trees is essential for algorithms involving hierarchical data, search operations, and many real-world applications like file systems, databases, and compilers.