The root is at the bottom. There are 2^3+1 terminals, including the root. The approximate Steiner tree is in blue. The angles are exactly, 120, 120-epsilon, 120+epsilon. The Steiner minimal tree for the same topology, in red, is constructed using the Melzak algorithm. There is a slider with which you can change epsilon.
Now there is a construction of a broken path which has the same length as the approximate tree. The straight-line path is the length of the Steiner minimal tree.
Equal angles have been indicated. If you look inside one of the small circles, say the circumcircle of triangle , then you'll see that the angle has to be at most 30 degrees for the Melzak construction to work.
Another requirement is that this circle should not swallow the Steiner point , but I don't yet know how to specify that.