It seems only right that the first Tree of the Week should be the first tree that any Roman came across in the city's history. When Romulus was a child, cast adrift to drown with his brother Remus, they were washed up on the shores of the Tiber, at the site of a fig tree, which would come to be known as the Ruminal fig. Accounts differ as to where the tree came in the story.
In visual depictions, the tree is with the twins. From a Republican coin (left) from 137 BCE to one minted in the reign of Maximinus Thrax in the 3rd century CE, to a fresco in the House of Marcus Fabius Secundus in Pompeii, the tree is at the site of the twin's suckling and their rescue. But the literary accounts differ, and the tree is a later addition.
Livy tells us that the twins were washed up 'where the Ruminal fig tree is now'. Florus says similar, as does Plutarch, who quotes a much earlier Roman historian, Fabius Pictor. The trees that these three (four if you count Pictor) historians are writing about are all at the Palatine hill in Rome, at a site that would become known as the Lupercal. We do not know precisely where the Lupercal was, Augustus built his house on top of it when he became emperor, and (despite many attempts) nobody has been able to definitively say where this site was. But it definitely existed in the first century CE, and Ovid tells us all about the Ruminal fig tree that he knew of.
The Augustan poet tells us that:
There was a tree there. Only a shell of it remains. It's named after Rumina now, but it used to be called the Romulean fig.
The aged, weathered fig tree at the Lupercal is all but gone by the time that Ovid comes across it, and has had a couple of names in its time. Fig trees can live for hundreds of years, so it could be that this fig tree had stood since the time of Romulus, from the eighth century BCE to roughly 8 CE. That could then mean that the fig tree depicted on the Augustan Ara Pacis, in a very fragmentary relief, is the Ruminal fig tree, and the aged version contemporary to its construction between 13 and 9 BCE. But given the significant damage that the scene has sustained, we are not able to say too much about it with any degree of certainty.
Strictly speaking, the Augustan period is when the Ruminal fig tree's life ends. A decrepit monument to a bygone age, it had been kept standing perhaps for its cultural capital, perhaps because of some religious connotations. We have never found the Lupercal, and we have never found any remnant of a tree that was a fourth participant in the iconic image of the she-wolf nursing the twin founders of Rome. Fortunately for the Ruminal fig tree, it experienced something of a resurrection after its death on the Palatine...