Branching points detection in processed image

Use MorphologicalBranchPoints.

im = Binarize@Import["https://i.stack.imgur.com/O0AMj.png"]

skel = Pruning[Thinning[im], 20];

HighlightImage[skel, MorphologicalBranchPoints[skel]]

Another possibility is to use

skel1 = Pruning[Thinning[im, Method -> "MedialAxis"], 10];

as a start then smoothen the result using

skel2 = Thinning@Dilation[skel1, 5]

so that MorphologicalBranchPoints would not give false results.

HighlightImage[skel2, MorphologicalBranchPoints[skel2]]

I notice there's a faint pink background that seems like a natural boundary. I've highlighted it to showcase this:

We can extract this curve and use it as the original boundary:

im = Import["https://i.stack.imgur.com/7Ck2S.png"];
mask = FillingTransform[Thinning[Binarize[ColorReplace[im, White -> Black, .055], 0]], CornerNeighbors -> True]

And the simply call MorphologicalBranchPoints:

skel = Thinning[mask];
HighlightImage[skel, MorphologicalBranchPoints[skel], 1]