BinaryTree minimum possible height

Question

I have one question for which I don't know any good solution (and my solution doesn't satisfy constraints). Namely, on one of the interviews i got problem like this:

Determine the minimum possible height of the binary tree, if you could remove numNodes leaf nodes from the tree (where after removal of all leaf children of node n, the node n becomes the new leaf node, and can be also counted as leaf). You are given the function int minimumHeight(Node* root, int numNodes), where root is the pointer to the root of the tree, which should be completed, using any oop language.

Constraints are that you can't actually delete node from tree, and you can't use queue or stack, arraylist, list, hashmap, hashset, etc. (you cannot use any built-in c++ stl data structure, you can though make your own from scratch)
My solution used BFS where I stored for each node its level in tree using hashmap m, and created an array arr which contained all nodes, and then sorted it based on node levels in ascending order, and after that the function should return m.get(arr[arr.length-numNodes-2]). (i wrote code in java)
Without constraints I think this could work,but I guess it's not good enough.. Does someone maybe know is there some more elegant way to solve this? (without using unnecessary data structures, and maybe without any sorting) Thank you in advance!

Here if numNodes is 5, the minimum height you could obtain is 2,because first you must delete nodes with 5,11,4 or 2 value,and then you can delete nodes with values 9,6,etc. so in total you will have 5 possibly removed nodes, and 2 will be the minimum height of the tree.

Answer 1

I would leave a comment but I have less that 50 rep.

So, if I understand the question, DFS (or a way to parse the tree) is inevitable and all you need is an array (one cell for each level, plus one for the leaves) to count how many nodes are on each level. So given the numNodes you can start decrementing from the last (non-zero) element and check which would be the last non-zero element you will reach.

Also you don't actually need a stack, cause it is already implemented from the OS for you. So you just need to use a recursive algorithm passing each time the correct part of the array to the next call to calculate the nodes in the sub-tree. Now, all you need to know is the initial length of the tree. You run again the kind-of the same recursive algorithm but this time you are interested to know the total number of elements in the tree. Then the length of the array would be equal to the length of the total elements in the tree.

Time: O(n)

Space: O(n)

I'm not sure if I can write code here. I hope I was clear :)