Snip! Snip!

This tree branching of leaves is, in fact, an inverted version of Pascal's triangle, and with the condition of snipping one branch out everyday, the triangle will be adjusted as shown below:

As we can see, the number 1 on the far left can contribute only 1 new leaf to the right original Pascal's triangle due to the line is cut out (as the tree branch is trimmed). Therefore, from the first level, there are 1+1 leaves, and on the next level, the number of leaves will run as $2^{n-1}$ + 1 for any $n^{th}$ level (day n). For example, on day 3, there will be $2^{3-1}$ +1 = 4+1 = 5.

Hence, on day 17, there will be $2^{17-1}$ +1 = 65537 leaves.

Let $S_n$ denote the number of leaves on n $^\text{th}$ day.

We know, $S_n = 2S_{n-1}-1$

Using Recurrence Relation, $\begin{aligned}S_n-2S_{n-1}&=-1\\\Rightarrow S_n&=c2^n+\dfrac{-1}{1-2}\\&=c2^n+1\end{aligned}$

Also, $\begin{aligned}S_1 = 2 &= c\cdot2^1+1\\\Rightarrow c &= \frac12\end{aligned}$

Thus on 17 $^\text{th}$ day, number of leaves will be $\huge S_{17} = 2^{16}+1=65537$