Frequently asked rooted tree algorithms/problems

In computer science, rooted trees are a hierarchical data structure made up of nodes linked by edges. They begin with a single node called the root node. The root node then branches out into a network of connected nodes. Each node can have one or more child nodes, with edges defining the parent-child relationships. Trees are important in computer science because they can represent different types of hierarchical data, such as organizational charts, organizational hierarchies, and file systems. They are used in many algorithms, like searching and sorting, to organize the data efficiently.

Let’s understand the different concepts/terms used in the trees:

Types of trees#

There are some special types of trees. Every tree type is suitable for different use cases and scenarios. Some of these types are listed below:

• Binary trees: In a binary tree, each node can have up to two children, generally known as the left and right children. Binary trees are the basic structure for more complex trees and algorithms. They’re used in many applications, like parsing expressions and creating binary heaps.

• Binary heaps: A binary heap is a type of binary tree where all levels are filled except possibly the last, and all nodes are as far left as possible. It follows a specific rule called the heap property, which has two types:

  • Max-heap: In this structure, each parent node has a value that is greater than or equal to its children’s values. This means the largest value will always be at the root (the top).

  • Min-heap: In this version, each parent node has a value that is smaller than or equal to its children’s values. This means the smallest value will always be at the root.

• M-ary trees: In an m-ary tree, each node can have up to $m$ children instead of just two in binary trees. Each node can have 0 to $m$ children. The tree is arranged so that all levels, except maybe the last one, are fully filled, and all nodes are as left as possible.

• Binary search trees (BST): A BST is a special kind of binary tree where, for any node, all values in the left subtree are smaller than the node, and all values in the right subtree are larger than the node. This makes searching, inserting, and deleting data efficient. BSTs are commonly used in database indexing, storing data in memory, and handling dynamic sets.

• AVL trees: An AVL tree is a type of self-balancing BST where the height difference between any node’s left and right subtrees is at most one. This balance keeps operations like search, insertion, and deletion fast in $O(log\space n)$ time. The AVL trees are useful when quick lookups and updates are important, such as in memory management and certain database systems.

• Red-black trees: Another self-balancing BST, red-black trees ensure that the longest path from the root to any leaf node is no more than twice as long as the shortest path. This balance ensures $O(log\space n)$ time for insertions, deletions, and lookups. Red-black trees are widely used in real-world applications, like implementing associative arrays and solving computational geometry problems.

• B-trees: B-trees are balanced trees that extend binary search trees to allow more than two children per node. They’re perfect for systems that handle large blocks of data, such as databases and file systems. B-trees stay balanced by keeping all leaf nodes at the same depth, which improves performance for disk-based operations.

• Trie (prefix tree): A trie is a tree-like structure that stores strings or sequences of characters, making it easy to find and match prefixes. Each node in a trie represents a common prefix of the strings stored, and paths down the tree show the different strings. Tries are commonly used in autocomplete systems, spell checkers, and IP routing.

Importance of tree algorithms in interviews#

Tree algorithms are a key part of technical interviews because they solve various problems and help evaluate a candidate’s grasp of data structures. Trees are used in many real-world systems, like databases, file systems, and network routing. Interviewers often choose tree-based problems to test problem-solving skills, knowledge of recursion, and familiarity with complex data structures.

Tree problems require a solid understanding of working with non-linear data structures. Unlike arrays or linked lists, trees involve algorithms that can effectively traverse, manipulate, and analyze nodes within a hierarchy. Knowing tree algorithms well is crucial for anyone looking to do well in software engineering interviews because they show a strong foundation in core computer science concepts.

We will focus on binary search trees because they are commonly used in computer science problems and algorithms.

Binary search tree traversals#

Traversal is the process of visiting every node in a tree in a specific order. Understanding the different traversal methods is important because many tree problems depend on visiting nodes in a certain way. The three main types of binary tree traversal are in-order, pre-order, and post-order, each with its own uses and applications. We will take the following binary search tree as an example for all three types of traversal:

Let’s now discuss all the traversals one by one.

In-order traversal#

In-order traversal involves visiting the left subtree first, then the root node, and finally the right subtree. This method is especially useful in BST because it visits the nodes in ascending order. During an in-order traversal, the nodes would be visited in the following order:

This sequence reflects the in-order property of a BST, where nodes are visited in ascending numerical order. In-order traversal is mainly used in BSTs to retrieve elements in sorted order. This in-order traversal is also useful in algorithms that need to process nodes in a sorted sequence, such as validating a BST to ensure it’s properly ordered.

Pre-order traversal#

Pre-order traversal involves visiting the root node first, followed by the left subtree and then the right subtree. During the pre-order traversal of the tree given above, the nodes would be visited in the following order:

This method is particularly useful for tasks that require processing the root node before examining its children. Pre-order traversal is often used to create a copy of the tree or to print the tree’s structure.

Post-order traversal#

Post-order traversal involves visiting the nodes in this order: first, the left subtree, then the right subtree, and finally, the root node. During the post-order traversal of the above tree, the nodes would be visited in the following order:

This method is particularly useful when you need to process child nodes before their parent, such as in tree deletion operations where a node should be deleted only after all its children have been removed.

Implementation#

The binary search tree traversals are commonly implemented using a recursive approach, but they can also be implemented with an iterative approach by using the stack.

Time and space complexity#

The time complexity of any tree traversal is $O(n)$ where $n$ is the number of elements in the tree. The space complexity is $O(h)$, where $h$ is the height of the tree.

Binary search tree operations#

Let's discuss the different types of operations we can perform in a binary search tree, like searching, insertion, and deletion.

Searching#

Searching in a BST utilizes the tree’s ordered structure to efficiently find a target value. The search starts at the root, where the algorithm compares the target value with the current node’s value. Depending on the result, it either moves to the left or right subtree. This process continues until the target value is found or a leaf node is reached.

Let’s try to find a value in a binary search tree using the help of an illustration:

Insertion#

Insertion in a BST involves finding the right spot for the new node while keeping the BST’s order intact. The process begins at the root, where the algorithm compares the new value with the current node’s value. If the new value is smaller, it moves to the left child; if it’s larger, it moves to the right child. This continues until an empty spot is found, where the new node can be inserted.

Let’s try to insert a value in a binary search tree using the help of an illustration:

Deletion#

Deleting a node from a BST requires careful handling to maintain the tree’s structure and properties. The process varies based on the node’s number of children:

• Leaf node (no children): If the node is a leaf node, it can simply be removed without any further changes to the tree.

• One child: If a node has one child, the node is removed, and its child is linked directly to the node’s parent, keeping the tree connected.

• Two children: The node is replaced by either its in-order successor (the smallest node in its right subtree) or its in-order predecessor (the largest node in its left subtree). This ensures the BST’s order is preserved.

Let’s try to delete a value with two children from a binary search tree using the help of an illustration:

Tree depth and height calculation#

Understanding the concepts of tree depth and height is important for many tree-related algorithms and operations. Although both concepts are related, they are different conceptually. Let’s understand the difference between these concepts.

• Depth of a node: Depth is the distance from the root node to the given node, i.e., it is the number of edges from the root to that node. The root node’s depth is always zero because there are no edges to traverse to reach it.

• Height of a node: The height of a node is the number of edges on the longest path from that node to any leaf node in its subtree. The height of the entire tree is the height of the root node, which represents the longest path from the root to any leaf node. In other words, the height of a tree is the maximum depth of any node within it.

Let’s see an example of the depth and height of the nodes of a BST.

Recursive and iterative methods#

We can find a tree’s height and depth (of each node) using recursive or iterative methods.

Let’s discuss both ways one by one.

Calculating height by using the recursive method#

The height of a BST can be calculated by recursively computing the height of each subtree. In the end, we return the maximum of these heights plus one (we added one to account for the current node).

Let’s see the code to calculate the height of a BST by using the recursive method:

Here’s a summarized explanation of the recursive height calculation code:

• The recursive method calculates the height of a binary search tree (BST) by exploring each subtree.

• If a node is None, it returns -1, indicating an empty subtree.

• For each node, the function recursively computes the height of its left and right children.

• The height of the current node is determined as the maximum of the left and right subtree heights plus 1 (to account for the current node).

• The process continues from the leaf nodes back up to the root, eventually determining the height of the entire tree.

This approach breaks the problem into smaller subproblems (recursive calls) that are later combined to compute the overall tree height.

Calculating depth by using the recursive method#

The recursive approach to finding a node’s depth in a BST traces the path from the root to the node, incrementing a counter with each step.

Let’s see the code to calculate the depth of a BST node by using the recursive method:

Here’s a summarized explanation of the recursive depth calculation code:

• The recursive method calculates the depth of a node in a binary search tree (BST) by tracing the path from the root to the target node.

• Starting from the root, if the current node is None, it returns -1, indicating the target node is not found.

• If the current node’s value matches the target, it returns the current depth.

• If the target is smaller than the current node’s value, it recursively searches the left subtree while increasing the depth by 1.

• If the target is larger, it recursively searches the right subtree, incrementing the depth by 1.

• This process continues until the target node is found or a None node is reached, returning the depth of the target node if found.

This approach breaks down the search into smaller steps, incrementally updating the depth at each level.

Calculating height by using the iterative method#

We can use a level-order traversal to count the levels of the tree, which corresponds to its height.

Let’s see the code to calculate the height of a BST by using the iterative method:

Here’s a summarized explanation of the iterative height calculation code:

• The iterative method calculates the height of a binary search tree (BST) using level-order traversal.

• If the tree is empty (i.e., the root is None), it returns -1, indicating no height.

• A queue (using deque for efficiency) is initialized with the root node to facilitate level-order traversal.

• A loop processes each level of the tree:

  • It determines the number of nodes at the current level by checking the queue’s length.

  • It dequeues each node at that level, checks for its left and right children, and enqueues them if they exist.

• After processing all nodes at the current level, the height counter is incremented.

• This process continues until all levels of the tree are traversed and the final height of the tree is returned.

This approach effectively counts the number of levels in the tree by systematically exploring each level before moving to the next, resulting in a clear and accurate measurement of the tree’s height.

Calculating depth by using the iterative method#

We can use an iterative approach that starts at the given root and counts the number of steps required to reach the required node.

Let’s see the code to calculate the depth of a BST node by using the iterative method:

Here’s a summarized explanation of the iterative depth calculation code:

• The iterative method calculates the depth of a node in a binary search tree (BST) by traversing from the root to the target node.

• A depth counter is initialized to 0, and a current pointer is set to the root.

• A loop runs until the current node is None:

  • If the current node’s value matches the target, the current depth is returned.

  • If the target value is less than the current node’s value, the search continues in the left subtree by updating current to its left child.

  • If the target value is greater, it moves to the right subtree by updating current to its right child.

• The depth counter is incremented with each step down the tree.

• If the loop exits without finding the target node, -1 is returned to indicate that the node is not present in the tree.

This approach effectively counts the steps taken from the root to the target node, providing an efficient way to determine its depth within the BST.

Time and space complexities#

Let’s break the time and space complexities for finding the depth and height of a binary tree using both recursive and iterative methods:

Recursive approach#

• Time complexity:

  • Height: To find the height of a tree, we need to visit each node once, so it takes $O(n)$ time, where $n$ is the total number of nodes.

  • Depth: Finding the depth of a node depends on how deep the tree is. In the worst case, if the tree is very deep, it can also take $O(n)$ time.

• Space complexity:

  • Height: The amount of memory used is based on the depth of the recursion, which is $O(h)$, where $h$ is the height of the tree.

  • Depth: Finding the depth also uses $O(h)$ space because of the recursion stack.

Iterative approach#

• Time complexity:

  • Height: Using level-order traversal (BFS) to find the height means visiting each node once, so the time complexity is $O(n)$.

  • Depth: Finding the depth of a node iteratively takes $O(h)$ time, where $h$ is the height of the tree.

• Space complexity:

  • Height: The space required is based on the widest level of the tree, which can be $O(n)$ in the worst case if the tree is very wide.

  • Depth: Finding the depth iteratively uses constant space, $O(1)$, as it only requires a few variables.

Frequently asked tree problems#

Trees are key data structures for representing hierarchical relationships in fields like computer science and databases. Mastering common tree problems is necessary for improving algorithms and solving complex challenges. Let’s discuss the overview of frequently encountered tree problems, including their significance, time and space complexities, and important concepts.

Next steps#

Now that you’ve mastered tree algorithms dive deeper into other essential data structures and algorithms through these helpful blogs:

• Master Algorithms with Python for Coding Interviews

• Top data structures and algorithms every developer must know

• Ace the top 15 Java algorithm questions for coding interviews

To further support your learning journey, explore Educative’s wide range of courses designed to enhance your algorithm skills:

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