from your question it doesn’t get clear if you need these points inside a java plugin or simply as a readout.
Here would be a possibility to get them via a macro. might not be the most elegant way but works.
The only drawback is, that it sometimes gives you a 1-pixel shift of the point depending on the orientation of the splitting line. but it might help to get started as an idea.
originalImage = getTitle();
run("Duplicate...", "title=[" + originalImage + "_ws]");
watershededImage = getTitle();
imageCalculator("Difference", originalImage, watershededImage);
run("Analyze Skeleton (2D/3D)", "prune=none show");
IJ.renameResults("Branch information", "Results");