The Baptism of Christ

We looked closely at the painting The Baptism of Christ by painter Piero della Francesca in class. While observing the painting, we were asked to begin writing things that we noticed in the painting. Immediately, my eyes were directed towards the Christ figure being baptised in the central region of the piece. Obviously this is the main focus of the painting based on both the location and the coloration. Further, I noticed the hovering dove, stripping man in the background, and the three angels looking onto the occurring baptism.
After class, I was still curious about this painting and it's hidden meaning. Through some web browsing, I came upon this website: http://www.independent.co.uk/arts. Reading their critique, the writers pointed of most of what we did in class. However, the one thing nobody thought about was the largely dominant tree in the upper part of the painting. The tree is very large and arches with the arch in the frame of the picture. Further, critics have identified the tree as a walnut tree, which is said to be a symbol of Christ's crucifixion. The walnut tree is also described as having guardian power in the scene.
Had I not further searched about this painting, I would definitely not have even paid any attention to the tree (which is normal according to the article). This is only one more detail of the painting that contributes to our understanding of its meaning.