From this short description of the data structure, we can already conclude that a Segment Tree only requires a linear number of vertices.

The first level of the tree contains a single node (the root), the second level will contain two vertices, in the third it will contain four vertices, until the number of vertices reaches $n$.

Thus the number of vertices in the worst case can be estimated by the sum $1 + 2 + 4 + \dots + 2^{\lceil\log_2 n\rceil}= 2^{\lceil\log_2 n\rceil + 1} \lt 4n$.

Thus the number of vertices in the worst case can be estimated by the sum $1 + 2 + 4 + \dots + 2^{\lceil\log_2 n\rceil}\lt 2^{\lceil\log_2 n\rceil + 1} \lt 4n$.

It is worth noting that whenever $n$ is not a power of two, not all levels of the Segment Tree will be completely filled.

We can see that behavior in the image.