Let us define a quasi-complete binary tree as a rooted binary whose nodes have all two children except at most those of the penultimate level, which can have either one or two children.
I read that if such a tree has $n$ nodes, then its depth is $\lfloor\log_2n\rfloor$, which is an interesting statement that I would like to understand.
I have tried to prove it to myself by using induction and I do see that it holds for $n=2$ (and $3$), but the floor function together with the logarithm presents some difficulties to me in the use of induction. Could anybody prove why the depth of such a tree is $\lfloor\log_2n\rfloor$? I thank you very much!