Motivated by Continuum image of line is chainable?

A planar continuum $X$ is a compact, connected subset of the plane. It is linear if there is a continuous bijection from $[0,1)$ onto $X$, for example a Warsaw circle. It is triodic if it contains a simple triod, i.e. three closed arcs all attached at an end point (3-pointed asterisk). It is tree-like if there is a sequence of surjections $f_n : X \rightarrow T_n$ onto finite trees such that max$[\text{diam}(f_n^{-1}(x))]$ goes to zero.

I strongly suspect no such object exists. It seems like the 'being tree-like' happens after finite time and then the tail will create a loop in the plane when it comes back around to start converging back onto itself. The inclusion of a triod then seems like it will cause impossibilities for the bijectiveness criterion of linearity.

Any thoughts? I found very little overlap in the investigation of linear continua and treelike continua.