Immediately following שירת הים we find the mysterious story in which בני ישראל, parched after a three day trek out of Egypt without water, come upon water that they are unable to drink. ה' shows משה a tree which, when tossed into the water, sweetens it for all to enjoy. The narrative is rich with symbolism and mystery, begging for a midrashic reading.
We are aware that water is used by חז"ל as a metaphor for תורה - just as water sustains physical life so does תורה sustain spiritual life. Perhaps then, this incident can be read as the search of בני ישראל for תורה. After all, isn't that what ה' essentially told משה was the central theme of יציאת מצרים (שמות ג:יב), which משה proceeded to tell the people (שם ד:ל) and repeated numerous times to פרעה (ה:א, ז:כו, ח:טז ועוד)? In essence, בני ישראל were beginning to think that they had been sold a bill of goods. They were promised Divine teachings if only they would follow the leader into the desert - yet here they went three days into the desert and there was no revelation of Divine knowledge.
This suggestion may be quaint and heartwarming, yet its thrust isn't felt till we investigate the rest of the incident. For בני ישראל actually do find water, yet it is bitter and undrinkable. Interestingly, ה' "teaches" משה a tree - ויורהו ה' עץ. It does not say ויראהו - ה' "showed" him, rather, ויורהו - ה' taught him. Somehow, it was the knowledge of the tree that turned the bitter into the sweet. According to tradition, בני ישראל were taught three מצוות at this juncture - שבת, כיבוד אב ואם, and the requirement to establish a justice system (סנהדרין נו: - see רש"י on שמות טו:כה for a variant version), following the implication in the text that there was teaching at this place. שם שם לו חוק ומשפט - it was there that they were taught laws and statutes (טו:כה). Apparently, the focus of their experience revolves around learning.
One piece of the puzzle still remains. What could be meant by the notion that ה' "taught" משה a tree, and how would teaching משה that tree resolve the complaints of בני ישראל? Following the analogy developed earlier, it could be suggested that there are two types of learning. One type of learning is dry and tasteless, uninviting and uninteresting, leaving one with a "bitter taste" in their mouth. Such learning doesn't inspire further learning; it is rote and meaningless. This was the full extent of the experience of the Jews prior to this incident. They were issued commands and given instructions to follow. They did not understand those rules - their job was simply to obey. (Perhaps this is what רש"י intended when he suggested that one of those rules was פרה אדומה, whose meaning is hidden, rather than כיבוד אב ואם, whose import is fairly apparent.)
At מרה, ה' taught משה that there was another way to teach - one that would excite, inspire and motivate, one that would leave the students tantalized with a desire to come back for more. משה needed to learn this from ה'. It was only after משה learned this that the bitter waters - the bitter learning - could be transformed into a sweet experience of understanding and immersion in תורה.
And what was the tree? I offer two possibilities. One, it was the tree of knowledge, from which we tasted in the beginning and whose impact has been bittersweet throughout the ages. Two, it was the עץ חיים היא למחזיקים בה - the tree of life to all who seize it. משה taught his people that the תורה is not merely a book of laws, requirements, duties, obligations, prohibitions and restrictions. It is a guide to all of life, and the very lifeblood of our spiritual experience.
Are there other trees in תנ"ך? Yes! And maybe it was one of those. Perhaps you can write the next line of this מדרש.