## Introduction

Vulnerability curves (VCs) have been measured in many hundreds of species since the first VCs were published (Choat *et al*. 2012). A sizable majority of them seem to fit the Weibull cumulative distribution function (CDF):

where *PLC* is percent loss of conductivity, *T* is tension, and *b* and *c* are Weibull constants. The Weibull CDF equation is preferred because it fits a wide range of curve shapes from S-curves to R-curves (Sperry *et al*. 2012).

However, we have seen a growing number of VCs that seem recalcitrant, that is, do not appear to follow the ‘Weibull rule’. Some recalcitrant VCs might be the result of flawed measurement technique or just measurement error, but we started to think that some recalcitrant VCs might be explained by structural features of the xylem of ‘recalcitrant species’. Usually VCs are characterized by just a few points (4–10) of *PLC* versus *T* needed to induce the *PLC* (see Fig. 1, solid symbols). It is difficult to determine the exact shape of the VCs when only a few points are measured. So, in this paper, we characterize VCs with 20–40 points, which can be carried out quickly in a Cochard cavitron (Cai *et al*. 2010a), with a high degree of accuracy quantified in terms of low standard errors of the *PLC* and *T*-values (see Fig. 1, open symbols).

In this paper, we demonstrate that the recalcitrant VCs measured in two species can be characterized accurately by a double Weibull equation. In one case, the double Weibull consists of one S- and one R-shape curve with closely spaced ‘*P*_{50} values’ on the T-axis. In the other case, the double Weibull is two S-curves that are widely separated on the T-axis. In the latter case, we hypothesized that the two S-curves correspond structurally to different conduit types in the stems, that is, either vessels of different size class (Cai & Tyree 2010) or the differences in vessel-to-vessel connections via pits or maybe vessels versus fibre tracheids.

In this paper, we use ‘fibres’ and fibre tracheids synonymously. Hence, ‘fibres’ have pit pathways for water flow between fibres or between fibres and vessels. In contrast, fibres without pits will be explicitly called libriform fibres. Carlquist (1984) reported that woody species with solitary vessels tend to be surrounded by water conducting ground tissue (fibre tracheids). Hence, one objective of this paper was to test the possible function of fibres and whether fibres might have a role in explaining dual Weibull curves. We used dye staining techniques, hydraulic measurements and quantitative anatomy measurements to test the hypotheses that fibres might form fibre bridges between adjacent vessels and that this structure might account for the dual Weibull curves.