Suppose we have a tree graph of order n. If we number the chains and branches like we do for alkanes, it seems that the max possible length of a branch is $\left\lfloor\frac{3n}7\right\rfloor$ if we expand on the tree graph and the main chain cannot be longer than n. (Note, branch is has less than or equal order of the main chain).
Example: Given a tree graph of order 3, the maximum length of a branch by increasing order is $\left\lfloor\frac{3\cdot3}7\right\rfloor=1$. This gives us the star graph of order 4.
I verified this by hand for $1\le n\le20$. I don't know if this is true or even how to attempt to prove it. I could not find anything on OEIS or Google.
Your formula is wrong for $n=9$. It predicts a maximum branch length of 3 whereas 5-butylnonane has a branch length of 4.
The correct formula is $\left\lfloor\frac{n-1}2\right\rfloor$. To maximise the length of a branch for a given main chain of length $n$, we should start the branch as close to the centre of the main chain; if the branch starts $k$ carbons from an end, any branch of longer than $k$ carbons will no longer be a branch but instead be part of the main chain. The maximum value of $k$, and thus the longest possible branch, are given by the formula I wrote above.