Fig. 1 Schematic diagram of Bayesian time tree construction (modified from: Bromham et al, 2018). The parameter setting of evolutionary models is the key step of constructing phylogenetic tree based on Bayesian method, which is composed of substitution model and tree model. The substitution model includes base substitution model (including base conversion rate rij and base frequency πi as parameters), site heterogeneity of rate (such as Γ distribution) and proportion of invariant sites p(inv). The tree model can be decomposed into two components: structure and branch lengths. The structure is determined by tree priors (in this case the birth and death process model), which includes three parameters: species generation rate (λ), species extinction rate (μt) and sampling frequency (ρ). In the “two-step” method, the structure can also come from the input of the tree file. The branch length is determined by the branch duration and the rate distribution across branches. The branch duration is influenced by tree priors and node age calibration. The rate distribution is recognized as clock models, which determine the distribution pattern of evolution rate on different branches. The data set is applied to the evolutionary model to generate posteriors of each parameter and the time tree.

Fig. 2 Schematic diagram of different molecular clock models (modified from Ho & Duchêne, 2014). The six time trees in this figure have the same structure, but the branch length varies greatly due to different selection of molecular clock models. (a) The time tree without applying molecular clock model. The scale bar indicates one time unit. (b) In strict molecular clock model, all branches have the same rate. (c) Local multi-rate molecular clock allows a certain amount of rate and sets the rate according to the clustering situation of topology. The related branches have the same or similar rate. (d) Discrete multi-rate molecular clock, which also allows a certain amount of rate, does not consider the topological clustering. (e) The autocorrelated relaxed molecular clock allows the existence of a rate at most equal to the number of branches, and the rate dependence of adjacent branches. (f) The uncorrelated relaxed molecular clock, without any restrictions on the number and distribution pattern of rates, is the most relaxed and free model setting.

Fig. 3 Schematic diagram of some terms in phylogenetic tree. The age of the crown node in the figure represents the node of the nearest common ancestor of all species in group A, while the stem node represents the node of the nearest common ancestor of the group A and its nearest related group. Taking the apomorphy-based method as an example, fossil A in the figure has synapomorphies with the extant species of group A, so it can be used as the minimum age constraint for the stem node of group A. To apply age constraint to the crown node of group A, it is necessary to obtain fossil records sharing synapomorphies with subgroups of group A, as indicated by the relationship of species Z, W and fossil B in the figure. The same is true of phylogenetic method.

Fig. 4 Three cases of fossil age uncertainty. (a) The present of “young but safe” and “old but risky” fossils in two dimensions: age of fossils and reliability. Most of the other fossils are distributed in the area surrounded by the two dotted lines. (b) The age of the oldest known fossil of a certain group tends to be close to the real divergence time of the group as the paleobiological excavation continues to expand. (c) With the development of fossil dating technology, the accuracy of fossil age has been improved.

Fig. 5 Probability distribution of node age. The maximum or minimum age constraint can be set on different nodes, and different probability distribution can be adopted. Where (a) shows two nodes with minimum and maximum age constraint, respectively, in phylogenetic tree; (b)-(f) emphasize the minimum age constraints in uniform, normal, gamma, lognormal and exponential distribution, while (g) and (h) represent strict and relaxed maximum age constraints on uniform distribution, respectively. In addition, the part highlighted by the dotted box in (c)-(f) is equivalent to the relaxed maximum age constraint. The soft maximum/minimum constraint in the figure is also called soft bound.