💡 Problem Formulation: Given a binary tree and a range [L, R], we are tasked with finding the number of nodes within this range. The binary tree is a rooted tree where each node contains an integer value. This article will explore various methods to solve this problem in Python. For example, if our binary tree nodes contain the values 1, 3, 5, and our specified range is [2, 4], the desired output should be 1, as only the node with the value 3 falls within the range.
Method 1: Recursive Tree Traversal
This method involves performing a recursive traversal (such as in-order, pre-order, or post-order) of the binary tree. During the traversal, we’ll check if each node’s value falls within the range [L, R] and count it if it does. This is an efficient and a straightforward way to tally nodes that meet the given criteria.
Here’s an example:
class TreeNode:
def __init__(self, val=0, left=None, right=None):
self.val = val
self.left = left
self.right = right
def count_nodes_in_range(root, L, R):
if not root:
return 0
return (L <= root.val <= R) + count_nodes_in_range(root.left, L, R) + count_nodes_in_range(root.right, L, R)
# Example Tree
root = TreeNode(10, TreeNode(5), TreeNode(15))
print(count_nodes_in_range(root, 4, 12))Output: 2
This code snippet defines a binary tree and a function count_nodes_in_range which counts the nodes within a given range. It uses a simple recursive algorithm, adding 1 for each node within the range and summing them up as the recursion unwinds. The elegance of this method is in its simplicity, which makes it clear and easy to understand.
Method 2: Iterative Approach with Stack
In this method, we use an iterative approach to traverse the tree while avoiding recursion. We maintain a stack to keep track of the nodes to visit. As we visit each node, we check if it is within the range [L, R] and increase our count accordingly.
Here’s an example:
def count_nodes_in_range_iterative(root, L, R):
stack, count = [root], 0
while stack:
node = stack.pop()
if node:
if L <= node.val <= R:
count += 1
stack.append(node.right)
stack.append(node.left)
return count
# Using the previous tree structure for TreeNode
print(count_nodes_in_range_iterative(root, 4, 12))Output: 2
This code implements an iterative version of the tree traversal using a stack. The function count_nodes_in_range_iterative pushes nodes to the stack and pops them in a loop to mimic recursion. It increments the count when the node’s value is within the given range. This method often outperforms recursion by avoiding the overhead associated with recursive function calls.
Method 3: Using BFS (Breadth-First Search)
Breadth-First Search (BFS) is another method to traverse the tree, using a queue to explore nodes level by level. It’s a way to ensure that closer nodes are checked before moving to deeper parts of the tree. We increment our count when encountering nodes with values within the specified range.
Here’s an example:
from collections import deque
def count_nodes_in_range_bfs(root, L, R):
queue, count = deque([root]), 0
while queue:
node = queue.popleft()
if node:
if L <= node.val <= R:
count += 1
queue.append(node.left)
queue.append(node.right)
return count
print(count_nodes_in_range_bfs(root, 4, 12))Output: 2
The function count_nodes_in_range_bfs performs a BFS on the tree. By utilizing a queue, nodes are visited in order of their depth from the root. Just like previous methods, when a node falls within the range, the count is increased. This method is particularly useful for wider trees where we might need to check many nodes at a shallow depth before going deeper.
Method 4: Depth-Limited Search
In some cases, we may want to perform a depth-limited search, i.e. search up to a certain depth. This is a variant of DFS where we maintain a current depth and stop the search when a certain depth limit is reached.
Here’s an example:
def count_nodes_in_range_dls(root, L, R, depth, limit):
if not root or depth > limit:
return 0
return (L <= root.val <= R) + count_nodes_in_range_dls(root.left, L, R, depth + 1, limit) + count_nodes_in_range_dls(root.right, L, R, depth + 1, limit)
print(count_nodes_in_range_dls(root, 4, 12, 0, 2))Output: 2
This code defines a function count_nodes_in_range_dls which is a modified DFS that only counts nodes within the range that are not deeper than a specified limit. The depth limit helps to restrict the search space, which can be useful when the range of interest is known to be within a certain distance from the root.
Bonus One-Liner Method 5: Using List Comprehensions and Generators
For those who love concise Pythonic solutions, list comprehensions combined with a generator function for tree traversal can be used to create a compact one-liner solution.
Here’s an example:
def count_nodes_gen(root):
if root:
yield root.val
yield from count_nodes_gen(root.left)
yield from count_nodes_gen(root.right)
print(sum(1 for v in count_nodes_gen(root) if L <= v <= R))Output: 2
This snippet uses a generator function count_nodes_gen to yield all node values in the tree and then uses a concise list comprehension to count how many of those values fall within the range. This is the epitome of Python’s ability to write compact and readable code.
Summary/Discussion
- Method 1: Recursive Tree Traversal. Simple and straightforward. The recursive stack may become a problem with very deep trees.
- Method 2: Iterative Approach with Stack. Avoids the drawbacks of recursive stack overflow. Manually manages a stack which can be less readable than recursion.
- Method 3: Using BFS. Useful for wide trees with many nodes at each depth. Not as memory-efficient for deep, sparse trees.
- Method 4: Depth-Limited Search. Offers control over the depth of traversal. May not be suitable when the depth of relevant nodes is unknown.
- Bonus Method 5: Using List Comprehensions and Generators. Pythonic and concise. May not be as obvious for beginners and isn’t necessarily performance optimal.