Due to the weight that plants can present, it is important to consider what could actually be planted on the bridge. For example as a general rule of thumb, a tree's roots can spread as wide as the tree is tall, as wide as 20ft!
Bamboo
Bamboo is a very realistic option for the bridge. It can grow fast. It can also be controlled easily too, growing as dense as desired. The height is easy to maintain as well. Another factor taken into consideration is the fact the canal area can act as a funnel for wind. If great trees were placed onto the bridge, they would catch the wind and the bridge would begin to sway. Being made from timber and having to support such vast weight, the swaying could very easily cause a structural failure.
Vines
Vines are brilliant for adding a bit of a natural aspect to almost any vertical option. For example, the handrails could be designed to encourage grape vines to crawl up them. As well being an attractive option, the fact that grapes are growing on the bridge amongst the public would be a drawing factor for visitors.
Hops
Hops are very light. They also grow similarly to vines in that they can crawl up things, however they can appear much more densely. They are a solid green and would provide an excellent base plant for the bridge, providing an interesting height for people to walk amongst. They also have a good, strong link with some local and well established buildings. The two most prominent being Ye Olde Trip To Jerusalem (a 4 minute walk away) and the Castle Rock Brewery (a 10 minute walk away). The possibility of working with them could be an interesting one.
Sources
http://www.deeproot.com/blog/blog-entries/how-deep-do-tree-roots-really-grow
http://www.bamboogarden.com/Grove%20thinning.htm#edible%20shoots
http://www.bamboogarden.com/Bamboo%20growth%20rate.htm
http://www.bamboogarden.com/barrier%20installation.htm
http://www.bamboogarden.com/FAQ%20general.htm
http://www.bamboogarden.com/care.htm
http://www.midwestsupplies.com/media/downloads/426/ive_got_my_rhizomes_now_what_2.pdf
http://www.deeproot.com/blog/blog-entries/how-deep-do-tree-roots-really-grow
http://www.bamboogarden.com/Grove%20thinning.htm#edible%20shoots
http://www.bamboogarden.com/Bamboo%20growth%20rate.htm
http://www.bamboogarden.com/barrier%20installation.htm
http://www.bamboogarden.com/FAQ%20general.htm
http://www.bamboogarden.com/care.htm
http://www.midwestsupplies.com/media/downloads/426/ive_got_my_rhizomes_now_what_2.pdf