BRANCHES AND LEAF-NUMBER. 



113 



Calculating from this table the mean and median positions of second- 

 ary branches, we get the results set forth in table 58. 



Table 58. — Constants for position of secondary branches. Totals for series. 



From these data we see that: 



(1) The average position of origin of secondary branches is at roughly 

 the 7th to 9th node from the proximal end of primary branches. 



(2) Fifty per cent of all secondary branches originate from the 

 seventh {ca. ) or more distal nodes of primaries. 



(3) There is very close agreement between the different series in 

 respect to the point of origin of secondaries. The mean for the combined 

 Carp Lake material agrees to the third place of figures with that for 

 the Ann Arbor material (Series IV). 



From these results it is clear that branches (a) do not occur with 

 equal frequency at all nodes, nor (6) as will be seen by comparison with 

 tables 26 and 27 (pp. 50, 51), can the positional distributions of branch 

 origins be regarded as random samples of the positional distributions of 

 "whorls in general." Instead it is found that the branches are so dis- 

 tributed in their points of origin that the great majority occur beyond 

 the first few proximal nodes. But it has been seen in what has gone 

 before that (a) taking all whorls on branches together there are in the 

 total more first whorls than second, more second than third, and so on 

 (cf . tables 30, 31, and 32, pp. 59 and 60) ; and (6) that the mean number 

 of leaves per whorl is lowest for first whorls and increases as we go in a 

 distal direction. That is to say, the frequency distributions for ' whorls 

 in general" contain a higher proportion of whorls which, owing to the 

 operation of our first law of growth, have a small number of leaves than do 

 the distributions for whorls at which branches originate. Naturally, then, 

 we should expect the means to be higher for the latter than for the former 

 distributions. Again, we have seen that distal whorls are less variable 

 than proximal, and that further, the whorls at which branches originate 

 include a higher proportion of distal ones than do the distributions for 

 the entire plants. So we should expect them to be less variable. While 

 it is thus clear that the ordinary growth factors account for a consider- 

 able part of the results which have been found for the "branch-origin" 



