i can't help thinking that a flat table structure is better. you're going to have the same entity in multiple tables. I recall writing a bloodstock database about 20 years ago - it was a flat file then but worked well.
anyway, i think Clive was really asking what you mean by a "tree". are you looking for a graphical representation, or a straightforward collection of textual statements with links but no graphical rendition, or what? how many levels up and down do you want to show?
do you always start from a descendant and work into the ancestors or from a given horse, do you want the ancestors and the descendants?