Bone J skeletonize does not give the centerline skeleton

I am trying to use BoneJ skeletonize function to skeletonize my tree like structure in 3D volume. Here is a small example example.tif (6.4 MB) .
I convert the image to binary, then use BoneJ>skeletonize. What I expected was getting the centerline of the entire tree like strcture connected as skeleton.

But I got this Capture, which is not definitely centre line skeleton. I am not sure which mistake I did. I also tried fiji>plugin>skeleton>skeleton2D/3D. Gave me the same output. My actual data is bigger than the example, and when I run on the actual data, I also got the wired skeleton structure.
Can you please advise?

Thank you

It is. You have lots of little cavities inside the structure, which the skeletonisation is respecting. You need to fill the little holes and try again.

Hi Michael,
Thank you very much for advice. I now do Fill holes (under Binary) and try skeletonize again. I got the output like this which is skeletonized structure


But I noticed that there are some side branches espedically in the thick middle branch (shown with red lines). These side branches arise (quite perpendicular) from the center line of thick branch. I am not sure why these branches are there while original volume does not have. How can I remove those side branches?

Thank you very much.