Step 1: Understand the conditions
The total number of nodes is $N = 15$.
The condition for every node $M$ is that the sizes of its left and right subtrees, $size(L(M))$ and $size(R(M))$, must satisfy the inequality $|size(L(M)) - size(R(M))| \le 1$.
Step 2: Determine subtree sizes
For the root node, we have $size(L(N)) + size(R(N)) + 1 = 15$, which simplifies to $size(L(N)) + size(R(N)) = 14$.
Given the condition $|size(L(N)) - size(R(N))| \le 1$, the only possible integer solution is $size(L(N)) = 7$ and $size(R(N)) = 7$.
This logic applies recursively to all subtrees, forcing the tree to be a perfect binary tree.
Step 3: Calculate the height
The number of nodes $N$ in a perfect binary tree of height $h$ (where a single node has height 0) is given by the formula $N = 2^{h+1} - 1$.
Step 4: Solve for the height
We substitute the total number of nodes $N = 15$ into the formula and solve for $h$:
$15 = 2^{h+1} - 1$
$16 = 2^{h+1}$
$2^4 = 2^{h+1}$
$h+1 = 4$
$h = 3$
Answer: The maximum possible height of the tree under the given conditions is 3.
