You can thin the image with bwskel and then use bwmorph to find branchpoints. I think those should be accurate enough. If they aren't, then tell us why.
You could erase the branchpoints to segment the image into separate segments and then use bwlabel and ismember and find OR regionprops to get each individual line's coordinates. Then use polyfit to get the equation of each line. Then have a double nested for loop where find the exact points where each pair of lines cross.
Or you could threshold for white and find the centroid of each white polygon. Then get the boundary with bwboundaries and compute the distance of each boundary coordinate from the centroid. Then use findpeaks to find the coordinates of the vertices. See attached demos.
