Suppose, I want to grow $6$ perfectly shaped mango trees.At first I plant $6$ saplings. Suppose, I want to grow $6$ perfectly shaped mango trees.At first I plant $6$ saplings.On each day after that , I sprinkle auxin solution to turn each sapling into a big tree that day, and for each sapling, I succeed in turning it into a big mango tree that day with a probability of $\frac{1}{2}$.Once a sapling is turned into a tree,it remains a tree.The expected number of days it will take me to get $6$ perfectly sized trees is $\frac{m}{n}$. Find $m,n$ I really have little idea about expectation function. So, I will be glad if you help me out

Let $f(k)$ be the expected number of days it take for $k$ saplings to turn into $k$ trees. Clearly, $f(0)=0$. For $k>0$ we have $$\tag1f(k)=1+2^{-k}\sum_{i=0}^k{k\choose i}f(k-i) $$ Even without looking for a general, closed formula, you can use $(1)$ to determine $f(6)$ in a few steps.