The Height of subtree in Heap
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.