From Fig. 1 we notice that Mc7 (Lower anterior facial height) is the parent node of 3 variables, St1 (SNA angle), Ja4 (Lower Gonial angle), and Ri21 (Symphysis length). Mc7 is measured from a point close (anterior nasal spine) to one of the points that constitute the SNA angle (point A) and both points are part of the same structure (maxillary), so the modification of the first one could be accompanied of a modification of St1 as well. On the other hand, the relationship between Mc7, Ja4, and Ri21 is explained given that the three correspond to vertical measurements and the modification of one should be accompanied by the modification of the other two variables.

Ja4 is the parent node of Ja9 (Cranial base and Mandibular length ratio), a relationship for which we do not have a satisfactory biological explanation since Ja4 is a vertical measurement and Ja9 is a horizontal one. In turn, Ja9 is the parent node of Ja8 (Mandibular corpus length), which is explained given that Ja8 is one of the measurements that make up Ja9.

Ri21 is the parent node of Ri19 (Condylar height) and Ja5 (Anterior cranial base length), a relationship that does not have an acceptable biological explanation, except that, as they correspond to linear measurements, they are influenced by the volumetric proportionality that exists between the structures given the greater or lesser general size of the skull.

St1 is the parent node of Ja3 (Upper Gonial angle), a relationship that could be explained since both represent sagittal growth, St1 indicates sagittal position of the maxilla with respect to the skull and Ja3 horizontal projection of the mandible; Normally, both structures tend to grow proportionally in the sagittal direction. Ja3 is the parent node of Ri18 (Posterior height), which is explained by the fact that both measurements share a reference point (gonion). Ri18 is the parent node of Ri10 (Maxillary depth angle) and Ja6 (Posterior cranial base length), there is no biological explanation for the relationship between Ri18 and Ri10 since one corresponds to a sagittal measurement and the other is vertical and they are measured in different areas of the face. In the case of Ri18 and Ja6 they use different landmarks but both measure posterior height of the face, so a relationship between both variables is clearly explained.

Ri10 is the parent node of Ri13 (Anterior Cranial length), a relationship that can be explained since both measurements share a reference point (Nasion).

Ri13 is the parent node of Ri11 (Palatal plane angle) and Ja12 (Jarabak’s ratio), Ri13 and Ja12 share a reference point (Nasion) and all three correspond to vertical measurements, so the relationship between them is justifiable.

Ri11 is the parent node of Mc3 (Linear distance from point A to nasion perpendicular), which is explained because they share a reference point in the maxilla (Nasion) and the modification of this point would produce a change in both variables.

Ja12 (Jarabak’s ratio) is the parent node of 3 variables Ri15 (Mandibular corpus axis), Mc6 (Maxillary length), and Ja1 (Saddle angle). Regarding this relationship, Ja12 and Ri15 correspond to measures indicative of the magnitude of vertical growth, Ja1 is part of Ja12, and with Mc6 instead, it cannot be explained biologically in a satisfactory way.

Ja1 (Saddle angle) is the parent node of Ja2 (Articular angle), which is explained given that both are contiguous angles that tend to compensate each other, that is, the tendency is that if one angle increases, the other tends to decrease in post of maintaining the proportionality of the face.

Ri15 (Mandibular corpus axis) is the parent node of Ja7 (Ramus height) and Ri17 (Mandibular ramus position), a relationship that is explained by the fact that Ri15 and Ri17 share a reference point (Xi), and that the three measurements correspond to vertical variables.

Ri17 is the parent node of Ri12 (Cranial deflection), a relationship that is explained by the fact that both measurements contain the Porion-Orbitale line.

Mc6 (Maxillary length) is the parent node of Ja10 (Posterior facial height), Ar5 (Nasolabial angle), and Ja11 (Anterior facial height), a relationship that does not have an acceptable biological explanation, except for the volumetric proportionality that exists between the structures that contain the landmarks corresponding to MC6, Ja10, and Ja11, given the greater or lesser general size of the skull and that Ar5 can be influenced by Mc6 since the upper lip rests on the maxilla, although this relationship is not direct since it depends mainly on the sagittal position of the maxilla. Ar5 is the parent node of Ri9 (Maxillary height angle), which could be explained by the fact that, as in the previous case, the position of the upper lip can be modified given the position of the maxilla, although this relationship is not direct since Ri9 corresponds to a indicative measure of the vertical and not sagittal position of the maxilla. Ri9 is in turn the parent node of Ja13 (posterior cranial base ratio to ramus height), a relationship for which we do not have a satisfactory biological explanation since, although both are vertical measurements, they correspond to different areas of the face.

Ja10 (Posterior facial height) is the parent node of Ri16 (Articular cavity position: Porion to Ptv), however, there is no direct biological explanation for this relationship; Ri16 is in turn the parent node of Ri20 (Condylar neck length), although the condyle is in relation to the joint cavity, we did not find an explanation for the relationship between the sagittal position of the joint cavity (Ri16) and the length of the neck of the condyle; Ri20 is the parent node of Mc5 (mandibular length), a relationship that could be explained by the fact that Mc5 has a reference point in the condyle and this is related to the joint cavity and both are sagittal measurements.

When analyzing the importance of each variable shown in Fig. 2, we notice that the four variables that turned out to have the greatest discriminatory power are: Ja4 (Lower Gonial angle), Ja12 (Jarabak’s ratio), Mc7 (Lower anterior facial height), Mc3 (Linear distance from point A to nasion perpendicular). In particular, the first 3 variables are the measurements that account for the direction of vertical growth of the mandible, which is the main determinant in the pattern of facial growth and it is therefore logical that they appear as the most important. On the other hand, the Mc3 variable is indicative of the sagittal position of the maxilla with respect to the skull, which is not considered a determinant of the pattern of facial growth, however it could be related, since the rotation of the mandible generally in normally, it is accompanied by a rotation of the maxilla in the same direction and magnitude.

In the case of Fig. 3, it is observed that Mc3 (Linear distance from point A to nasion perpendicular) is the parent node of Ja4 (Lower Gonial angle), however, there is no direct biological explanation to explain this relationship since Mc3 is a sagittal measurement of the maxilla and Ja4 a vertical measurement of the mandible. In turn, Ja4 is the parent node of the variables Ja12 (Jarabak’s ratio) and Mc7 (Lower anterior facial height), which can be explained because the three variables correspond to measures indicative of vertical growth, so that when increasing or decreasing a of them, the others also increase or decrease respectively proportionally.