Problem Description

In this problem, we are given a binary tree where each node has an integer value. Alongside, we are given an integer k. The task is to find and count the nodes in the binary tree that are considered "great enough". A node is "great enough" if it meets the following criteria:

• The subtree rooted with this node has at least k nodes.
• The value of this node is greater than the value of at least k nodes in its subtree. We must explore the tree and count how many nodes satisfy these conditions.

Intuition

The intuition behind the solution is to perform a depth-first search (DFS) starting from the root node, to explore the entire tree. While exploring, we need to keep track of the nodes and their values in a way that enables us to check if a node is "great enough" according to the given conditions.

To check if a node is "great enough", we accumulate the greatest values we encounter in its subtree, including the node itself. We use a min-heap data structure, which allows us to efficiently keep track of the largest elements (since in Python's heapq the smallest elements come first, we push negatives of the values so that when we pop, we pop the smallest of the largest values).

Here's the key to our approach:

• Perform DFS from the root.
• For each node, merge the heaps of its left and right children to congregate their greatest values.
• Ensure our heap does not exceed k elements.
• Once merged, we compare the negative of the top of our min-heap (which is the kth largest value) with current node's value to determine if our node is "great enough".
• If it is, increment our counter ans.
• Finally, push the negative of the current node's value onto our heap (to include the current node in the subtree for future parent nodes) before returning the heap to its parent call.

The main advantages of this approach are:

• Space efficiency, as we carry a single heap up the tree while we are gathering subsets of values.
• Time efficiency, since heaps provide O(log k) pushing and popping, making the overall complexity better compared to sorting the values or other brute force methods that might seem tempting at first glance.

Solution Approach

The provided solution uses a Depth-First Search (DFS) algorithm in combination with a min-heap to solve the problem. Here is a step-by-step explanation of the implementation:

• A DFS dfs function is defined, which will be called recursively to explore the tree nodes. At each node, this function will return a min-heap containing the largest values (negated) from its subtree, with size at most k.

• Within this dfs function, when the base case is encountered (i.e., when a None node is encountered), an empty heap is returned, indicating that a leaf has been reached.

• When at a non-leaf node, DFS is called on both the left and right children, and their returned heaps are merged. It's important to note that we're using the push helper function to maintain the heap's size at no more than k.

• The push helper function works as follows: It pushes an element onto the heap and if the heap size is greater than k after the push, it pops the smallest element (which is represented as the largest negative value). This ensures that only the k largest (negated) elements remain.

• Back in the dfs function, after merging the heaps from the left and right subtrees, we check if the current node's value is "great enough". This check is done by ensuring that the heap size is exactly k and the top element of the heap (i.e., the smallest of the k largest negated values) is less than the negated value of the current node. If these conditions are met, it means the current node is greater than at least k nodes in its subtree, and we increment our answer ans.

• Finally, the current node's value (negated) is pushed onto the heap, and the heap is returned to be merged in the parent node's DFS call.

• After the DFS traversal completes, the counter ans holds the number of "great enough" nodes, and this value is returned as the solution to the problem.

Here is the pseudocode for the DFS and push function:

def dfs(node):
    if node is None:
        return new empty heap
    left_heap = dfs(node.left)
    right_heap = dfs(node.right)
    # Merge heaps
    for value in right_heap:
        push(left_heap, value)
    # Check if the current node is "great enough"
    if heap size is k and (-top of heap < node's value):
        ans += 1
    push(left_heap, -node's value)
    return left_heap

def push(heap, value):
    add value to heap
    if size of heap > k:
        remove the smallest element from heap

Notice that the dfs function's merge step iterates over the right heap and pushes its elements into the left heap. This merging avoids creating a new heap at each node, which saves memory and potentially decreases runtime due to fewer allocations.

Example Walkthrough

Let's use a small binary tree example and k = 2 to illustrate the solution approach.

Imagine a binary tree like this:

       5
      / \
     3   8
    / \   \
   1   4   10

In this tree, we want to count the number of "great enough" nodes, i.e., nodes with at least k nodes in their subtree and a value greater than at least k nodes in their subtree.

1. Begin DFS at the root node (5). Since it's not None, we proceed.

2. Perform DFS on the left child (3).

   • Left child (3) is not None, we perform DFS on its children.
     • Left child (1) has no children (both None), so it returns an empty heap.
     • Right child (4) has no children (both None), so it returns an empty heap.
   • Merge returned heaps from 1 and 4 and push negated values: left_heap contains [-4, -3].
   • Now, left_heap has 2 elements (size is k) and top of heap is -4, which is smaller than -3. So, node 3 is "great enough".
   • Increment ans to 1 and push -3 into the left_heap before returning it.

3. Perform DFS on the right child (8).

   • Right child (8) is not None, perform DFS on its left and right children.
     • It has no left child, so returns an empty heap.
     • Right child (10) has no children (both None), so it returns an empty heap.
   • Merge returned heaps and push negated values: right_heap contains [-10].
   • Heap size is less than k, so we cannot say node 8 is "great enough" yet.
   • Push -8 into the right_heap before returning it.

4. Back to the root now, left_heap contains [-4, -3] and right_heap contains [-10, -8]. Merge them.

   • After merging, the combined heap will have [-10, -8, -4, -3]. We need to maintain only the largest k values, so we pop off -3 to get left_heap as [-10, -8, -4].
   • The top of heap is now -4, which is smaller than -5. Thus, the root node (5) is "great enough".
   • Increment ans to 2. Push -5 into left_heap and return left_heap.

5. After DFS traversal, the ans is 2, representing the number of "great enough" nodes (nodes 3 and 5).

Thus, the output for our binary tree example with k = 2 is 2, meaning there are 2 nodes that are "great enough" by those criteria.

Solution Implementation

Time and Space Complexity

The time complexity of the given countGreatEnoughNodes function is O(n * log(n)), where n is the number of nodes in the binary tree. The function performs a depth-first search (DFS) on the tree, visiting each node once (O(n)). At each node, the push function potentially performs a heap push and pop operation in the priority queue. The length of the priority queue is capped at k, so each heap operation is O(log(k)). Since k can, at maximum, be as large as n (the total number of nodes), in the worst-case scenario, this becomes O(log(n)). When multiplied by the total number of nodes visited, we obtain O(n * log(n)).

The space complexity of the function is O(n). This is because the recursive DFS function call stack could potentially store every node in a path from the root to a leaf in the case of a skewed binary tree. In addition, the priority queue l also stores up to k elements, but since k is at most n, it does not exceed O(n) space. Thus, the overall space complexity is determined by the maximum between the height of the tree and the number length of the priority queue, which is O(n).