  We build on recent work of Yeats, Courtiel, and others involving connected
chord diagrams. We first derive from a Hopf-algebraic foundation a class of
tree-like functional equations and prove that they are solved by weighted
generating functions of two different subsets of weighted connected chord
diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle
subdiagrams. These equations generalize the classic specification for
increasing ordered trees and their solution uses a novel decomposition,
simplifying and generalizing previous results. The resulting tree perspective
on chord diagrams leads to new enumerative insights through the study of novel
diagram classes. We present a recursive bijection between connected
top-cycle-free diagrams with $n$ chords and triangulations of a disk with $n+1$
vertices, thereby counting the former. This connects to combinatorial maps,
Catalan intervals, and uniquely sorted permutations, leading to new conjectured
bijective relationships between diagram classes defined by forbidding graphical
subdiagrams and imposing connectedness properties and a variety of other rich
combinatorial objects. We conclude by exhibiting and studying a direct
bijection between diagrams of size $n$ with a single terminal chord and
diagrams of size $n-1$.
