Fixed point theorems for certain tree-like continua

It has been shown that not all tree-like continua have the fixed point property; i.e., if X is the inverse limit of trees with arbitrary bonding maps, then X may or may not have the fixed point property. We show that if the bonding maps satisfy certain conditions then the inverse limit space must have the fixed point property. More specifically, we show that weakly confluent maps of trees which throw the domain tree onto the image tree in a certain manner must be universal. In addition, we show that weakly confluent self maps of trees are universal. It follows that inverse limits of trees with weakly confluent bonding maps which satisfy the conditions mentioned above must have the fixed point property. Also, it follows that inverse limits on a single tree with weakly confluent bonding maps must have the fixed point property.