There are multiple things wrong with this, although I know you weren't aiming to exhaustive completeness. So my critique is directed at the philosophical argument, not you personally.
1. The tree issue is a false choice. Maybe I can just dig a hole and transplant the smaller one; perhaps I can modify my needs.
2a. From a callously practical point of view, the 50 year old doctor-professor will make a much larger and more satisfying meal for the hungry shark than the baby, which won't be much more than an appetizer.
Of course, we have to ask why we're putting the interest of humanity in general on such a pedestal, considering that society has yet to dispatch a rescue craft. Maybe we should keep the doctor around to treat the victims of future shark attacks or dehydration; in this sense 'society' looms only as large as our future life expectancy after the shark has gone. Then again, the doctor is heavier than the baby, and will consume a greater amount of food...
2b. Everyone else in this boat must be a Nobel prize winner or something if the successful medical researcher is the most disposable person after the baby. It's another false choice, in that it bolsters the hearer's ego while simultaneously limiting their choices - this exercise doesn't cater for the suggestion that one might sacrifice themselves to save both doctor and baby, for example - although the sad fact is that few people would, so in that sense it's accurately predicated on humans' general selfishness.
However, as a false choice it's dangerously seductive, and politicians work this one (or an inverted version) all the time: "We can't increase the debt (lifeboat)! Should we give a tax cut to the rich so they can create jobs (proven success) or waste it on giving welfare to bums (unknown outcome)?" or alternatively "My opponent is soft on crime. Are we going to get tough on criminals (proven bad) now or wait for some other innocent to die (unknown probability)?" You can construct endless variations on this, regardless of your political leanings: just stick with the formula of "Time is running out - do you want distasteful alternative A or repulsive alternative B?" and just tailor your argument to the known biases of whatever group you are addressing.
Underlying this argument is the implicit threat that if you don't make a choice, you'll be chosen instead. Republicans hint that not voting for the wealthy folks' interests could cause them to cut your job, Democrats murmur that the rich might find themselves sinners in the hands of an angry mob. Broad arguments that suggest joint stakes and responsibility (eg raise some taxes and cut some spending) tend to upset everyone, which is why they're only ever offered by politicians who are leaving office anyway.
2c. Value of the baby is unpredictable. Certainly if you take 100 babies and measure something (anything) about them now or in the future, you'll get a nice Gaussian distribution. But an individual baby's future is no more predictable than an individual roll of a dice. You are not any more likely to throw a 3 or a 4 than a 1 or a 6 or a 2 or a 5, even though the average of 100 throws will converge on a value of 3.5. Evaluating the probability in light of the doctor being a '6' is seductive, but only as long as you keep sidestepping the question of who else is in this boat that makes the doctor seem 2nd-most disposable. If the baby is the child of two world-class geniuses (neither of whom we would throw out before the doctor), then the probability of its future utility is high.
We could also wonder about what the average person will be capable of in 50 years' time thanks to technology, vs. the skilled person of today. Their relative utility to society then may be lower than that of the doctor to society today, but it might be larger in absolute terms. I am particularly wary of assuming the doctor's future utility over 25 years of natural life will exceed that of the average person's 50 years.
But the unpredictability of the individual is the more important factor. Suppose there are 100 lifeboats, none of which know about the others. If all 100 follow your advice and toss the babies, those with excellent future potential will be lost as surely as those of average or lesser potential.
2d. But the big issue here is the moral question, and here we have more rational responses than 'it's a baby, it feels so wrong!' Factors of weight, food consumption, shark appetite and life outcome probabilities aside, the basic argument against such a course is the defenseless nature of the baby; s/he cannot volunteer, acquiesce or even resist being thrown to the hungry shark the way the 50 year old doctor can, and this gives us an excellent reason to refuse the idea.
Is it because of a moral obligation towards defenseless beings? Sort of, but it's not rooted in any abstract philosophy. It's rooted in our self-interest: if we are in favor of sacrificing the defenseless, then then we are advertising our utility calculus to others in the boat. True, we are not sacrificing the baby because of its defenselessness, but we are not letting the fact mitigate against the utility function either. In a future round, we might throw out someone who's asleep because their wakeful utility is similarly imponderable; that increases the danger for all, since all need to sleep; nor can we exclude ourselves, since we began with the position that circumstances brook no delay.
Even if none of us sleep, or more sharks will arrive before we can, revealing our utility calculus allows everyone else to model it for themselves in relation to everyone else. Anyone expecting to be on the wrong end of a future comparison - the doctor/professor, for example - has an enormous incentive to throw us overboard first. Such a person can make a rather compelling argument that 'after me, you're next' to everyone whose assessed utility would be lower than our own; only those for whose survival we are willing to sacrifice ourselves have an incentive to let us remain. We have already demonstrated that we consider ourselves to be at least two steps away from being expendable simply by proposing the baby go overboard; on a practical level we might expect the mother to vote the interests of her baby, and so be similarly expendable to the extent that we consider her objections to increase the risk for the whole group.
On the other hand, if we argue for the survival of the baby, we are advertising our unwillingness to make decisions about unknowables and to defend the vulnerable, an attractive quality to everyone on the boat who is not fully assured of their own survival and social utility. Given the prevalence and success fo tit-for-tat strategies in iterated prisoner's dilemma games, it's probably in our interest to identify ourselves as cooperators rather than defectors. I say probably since there's always the possibility of being on a boat full of defectors...in which case it might be better to take one's chances with the shark, who is at least honest about his intentions.
If this seems a rather involved reply, it's because I've thought about this in more than purely hypothetical terms. A day after her birth, my wife was taken aboard a ship which was part of a refugee convoy of 8 vessels. There was a storm on the second or third night of the journey, and 6 of the boats sank, with no survivors. Besides the horrifying experience my in-laws endured, I'd say she outscores me on a social utility scale too.