This paper will be of interest to anyone who wonders which tree-like continua can be embedded in the plane. A special part of this problem, “Which simple-triod-like continua can be embedded in the plane?”, is an extention of an outstanding open question from 1972: Given an arc-like continuum \(X\) and an arbitrary point \(x\in X\), does there exist an embedding \(H\), of \(X\) in the plane, such that \(x\) is an accessible point [S. B. Nadler jun., in: Proc. Univ. Oklahoma Topol. Conf. 222–233 (1972; Zbl 0283.54017); S. B. Nadler jun. and J. Quinn, Embeddability and structure properties of real curves. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0236.54029)]. While this question remains open, the authors of the paper under review make some progress towards an affirmative answer. Since an arc-like continuum is homeomorphic to the inverse limit of an inverse system \(\{[-1,1],f_n\}\) with piece-wise linear bonding maps, the problem can be reduced to this situation, with the mappings normalized by \(f_n(0)=0\). The authors define the concept of a “radial departure” of a function \(f:[-1,1]\to [-1,1]\) to impose a condition on the bonding maps of the inverse limit which will ensure the existence of an embedding in the plane under which the point \((0,0,\dots)\) is accessible. To review the definition, for an arc-like continuum \(X\), a point \(x\in X\) and an embedding \(H\), of \(X\) in the plane, we say that \(x\) is accessible if there exists an arc \(A\) in the plane, such that \(A\bigcap H(X)=H(x)\). The work is a natural follow-up to [A. Anušić, Topology Appl. 292, Article ID 107519, 14 p. (2021; Zbl 1460.54010)], in which one of the authors answered affirmatively a question of Minc, whether a particular simple example of an arclike continuum, which is an inverse limit of an inverse sequence \(\{[0,1],f\}\) where all bonding maps are a specific linear function, can be embedded in the plane in such a way that the point \((1/2,1/2,\dots)\) is accessible. In their development of radial departure and its orientation, the authors use a lemma from [\textit{A. Anušić} et al., Topol. Proc. 56, 263--296 (2020; Zbl 1458.37025)], albeit with an alternative proof. The gist of this result lies in the assertion that the inverse limit of the inverse sequence \){[-1,1],f_n}\( has an embedding in the plane such that \)(0,0,…)\( is accessible, provided that for each \)n\( all radial departures of \)f_n\( have the same orientation (either positive or negative). The concept of radial departures of a piecewise linear function, and the associated finitely many ``contour points" allow the authors to describe the ``zig-zag'' behavior of such mappings using radial contour factorization. The main result of the paper proves the existence of the embedding of the inverse limit of the sequence \){[-1,1],f_n}\( in the plane for which \)(0,0,…)\( is accessible, under two conditions: 1) \)f_n\( and \)f_nοf_{(n+1)}\( have the same radial contour factor; 2) \)f_n\( has no contour twins. This last assumption can, hopefully, be removed from the future theorems, by the authors\)