In this Grasshopper definition, you can define a planar closed curve and a series of points for the branching base locations. use the Shortest walk plugin to find the nearest distance between the starting point and the branches. Finally, you can give it a thickness.
The ShortestWalk add-on for Grasshopper exposes one component which, given a network of curves and a list of lines, calculates the shortest route from line start point to line end points in a network. It is based on a topology calculator and the A* search algorithm.