The Height of subtree in Heap In order to find the recurrence function of The Height in Heap, the following figure is drawn.



Question 1: How can we compute the height if Right subtree in form of Log in base 3, and why do we have the height of Left subtree as log in base 3/2?

Question 2: it is stated that:
 "for a complete binary tree to have the maximum height the last level should be half full."
i don't grasp the logic behind this line.why?
You are asking several question. I will only answer one. If you start with a set of size $n$ and you repeatedly reduce it to at most $rn$ elements, then after $t$ steps, you will have at most $r^tn$ elements. If $r^tn &lt; 1$, then after $t$ steps you will have no elements. This happens for $t &gt; \log_{1/r} n$. This explains where $\log_{3/2} n$ and $\log_2 n$ come from.