  We introduce a new theory of plane $\mathbb R$-tree, to define plane ICRT
(inhomogeneous continuum random tree), and its looptree, fennec (a Gaussian
free field on the looptree), and snake. We prove that a.s. the looptree is
compact, and that a.s. the fennec and snake are continuous. We compute the
looptree's fractal dimensions, and the fennec and snake's H\"older exponent.
Alongside, we define a Gaussian free field on the ICRT, and prove a condition
for its continuity. In a companion paper , we prove that the looptrees,
fennecs, and snakes of trees with fixed degree sequence converge toward the
looptrees, fennecs and snakes of ICRT.
