**APPENDIX A: SOLUTION OF EQUATION 12 FOR FLOW UNDER TENSION**

We model the water pathway with the idealized vein doublet discussed above, where the

dimensions of the smaller vein-node pair are 1/*b*th the larger. Repeating eq 9:

1.94 10 -3 dv

4

*k*i =

(A1)

G2 (1 + *b*4 )

where *G *= 3.2 10 m3. The volume *v *of water re-entering the sample is Φ*V*, where *V*

(= 250 cc) is the sample volume. With *v *measured as 0.2 cc we take porosity as about 0.001.

The water re-entry pressure *P*Wn is given by eq 5d

*P*Wn = 0.1180/*d*v.

(A2)

For assumed values of *k*i and *b*, eq A1 yields the vein width *d*v. Eq A2 then provides the

re-entry pressure *P*Wn.

We solve eq 11, A1 and A2 simultaneously by trial and error, inputting values for *k*i and *b*

until the observed time is matched for a height of *z *= 0.043. This simplified scenario should