### Abstract

- Top of page
- Abstract
- 1 Material and methods
- 2 Results
- 3 Discussion
- 4 Conclusions
- Acknowledgments
- References
- Appendix

**Abstract ** We propose a simple statistical approach for using Dispersal–Vicariance Analysis (DIVA) software to infer biogeographic histories without fully bifurcating trees. In this approach, ancestral ranges are first optimized for a sample of Bayesian trees. The probability *P* of an ancestral range *r* at a node is then calculated as where *Y* is a node, and *F*(*r _{Y}*) is the frequency of range

*r*among all the optimal solutions resulting from DIVA optimization at node

*Y*,

*t*is one of

*n*topologies optimized, and

*Pt*is the probability of topology

*t*. Node

*Y*is a hypothesized ancestor shared by a specific crown lineage and the sister of that lineage “

*x*”, where

*x*may vary due to phylogenetic uncertainty (polytomies and nodes with posterior probability <100%). Using this method, the ancestral distribution at

*Y*can be estimated to provide inference of the geographic origins of the specific crown group of interest. This approach takes into account phylogenetic uncertainty as well as uncertainty from DIVA optimization. It is an extension of the previously described method called Bayes-DIVA, which pairs Bayesian phylogenetic analysis with biogeographic analysis using DIVA. Further, we show that the probability

*P*of an ancestral range at

*Y*calculated using this method does not equate to pp*

*F*(

*r*

_{Y}) on the Bayesian consensus tree when both variables are <100%, where pp is the posterior probability and

*F*(

*r*

_{Y}) is the frequency of range r for the node containing the specific crown group. We tested our DIVA-Bayes approach using

*Aesculus*L., which has major lineages unresolved as a polytomy. We inferred the most probable geographic origins of the five traditional sections of

*Aesculus*and of

*Aesculus californica*Nutt. and examined range subdivisions at parental nodes of these lineages. Additionally, we used the DIVA-Bayes data from

*Aesculus*to quantify the effects on biogeographic inference of including two wildcard fossil taxa in phylogenetic analysis. Our analysis resolved the geographic ranges of the parental nodes of the lineages of

*Aesculus*with moderate to high probabilities. The probabilities were greater than those estimated using the simple calculation of pp*

*F*(

*r*) at a statistically significant level for two of the six lineages. We also found that adding fossil wildcard taxa in phylogenetic analysis generally increased

_{y}*P*for ancestral ranges including the fossil's distribution area. The Δ

*P*was more dramatic for ranges that include the area of a wildcard fossil with a distribution area underrepresented among extant taxa. This indicates the importance of including fossils in biogeographic analysis. Exmination of range subdivision at the parental nodes revealed potential range evolution (extinction and dispersal events) along the stems of

*A. californica*and sect.

*Parryana*.