Adding curves to a 1D line tree defined by points and connectivity

Hello, thanks for reading this.

I was looking to create a curved tree network based off of an input of lines defined by points and connectivity. I looked a bit at the literature, and I think this problem has been solved many times over, but I'm having trouble because in the examples the lines are continuous. For me, I have n lines defined by points and connectivity.

Here is a snippet of code to give you an idea of how they look:

    ptMx = [  232.1498 -213.4505   94.4044
  232.1498 -213.4505   52.2680
  205.9872 -213.2559   25.8087
  259.1540 -213.6514   28.2015];

    faceMx = [1 2; 2 3; 2 4];

    radii   = [1.6; 1.6; 1.2; 1.1];
    colorMx = colormap(jet(size(radii,1)));
    colorMx = fliplr(colorMx);

    h = patch('Vertices', ptMx, 'Faces', faceMx, 'FaceVertexCData', colorMx, 'EdgeColor', 'interp');

And what I want to do is add something like a Centripetal Catmull–Rom spline to make the lines connecting each point curved.

Before I continue, I just wanted to ask for helpful advice from people who are much smarter than I am on this issue. Any ideas?