Tree - 4

The number of line-segments in the first image = $1$

The number of line-segments in the second image = $1 + 2$

The number of line-segments in the third image = $1 + 2 + 4$

The number of line-segments in the fourth image = $1 + 2 + 4 + 8 =$ $\boxed{15}$

A generalization: The number of line-segments in the $n^{th}$ image = $1 + 2 + 4 + \ldots + 2^{n-1} = 2^n - 1$ .

The number of line-segments $n$ of the $k^{th}$ image is given by:

$n = 2^k - 1$

For $k=4$ , $n=2^4-1 = \boxed{15}$