Automatically assigning attribute to line features

It is a long process but doable without programming:

dissolve your lines (no multiparts)
derive end points and delete overlapping
spatially join end points (1:m) and count number of lines connected to node:
buffer ones with count of 3 and intersect with original lines
dissolve intersect to multipart lines using buffer fid
derive ends of multipart lines and remove ones that intersect original nodes
convert points to line (close line) using relevant field to form a line:
create polygons from closed lines. This is a key, because polygons are made of clockwise points (!)
assign unique ID to each polygon and split it at vertices
compute their length, sort ascending by length and remove duplicates of polygon ID.